You can read the Symposium online in English at the Perseus Project and in French at Wikisource. C. D. C. Reeve has a good article on Plato's account of friendship at the SEP.
In the frame narrative:
Apollodorus is mentioned in the Apology and Phaedo, and has speaking parts in Xenophon's Memorabilia and Apology. He is represented by both Plato and Xenophon as being quite an emotional person.
Apollodorus is speaking with an anonymous companion.
Glaucon is Plato's brother. Apollodorus tells of a conversation he had with Glaucon in a subdialogue in the frame narrative.
In the main narrative:
Aristodemus of Cydathenaeum
Both Plato and Xenophon remark on his short stature; Xenophon gives him as an example of Socrates giving good advice to his close friends in Memorabilia 1.4.
Agathon of Athens
Agathon is mentioned in Protagoras as listening to Prodicus. He is a tragic poet who has want a great victory at a festival. The major victory party was the day before, but he is still celebrating with a more intimate dinner party for friends.
Pausanias of Cerameis
Pausanias is mentioned in Protagoras as listening to Prodicus. Pausanias is Agathon's lover, and several of the speeches are clearly designed to flatter the two.
Aristophanes of Cydathenaeum
Aristophanes is the only person given a full speech who was not at the sophist get-together in Protagoras.
Eryximachus of Athens
All three times Eryximachus is mentioned in Plato (Protagoras, Phaedrus, and here), he is mentioned as a friend and associate of Phaedrus. In Protagoras, he and Phaedrus are listening to Hippias.
Phaedrus of Myrrhinus
Phaedrus is the same Phaedrus as in the dialogue of the same name. In Protagoras he is mentioned as listening to Hippias.
Diotima of Mantinea
Diotima is not actually present, but she plays such a big role in Socrates' speech it seems appropriate to treat her as a character. She is a priestess, whom Socrates claims kept the great plague away for ten years by giving Athens good advice on sacrifices. There are a lot of Plato scholars who think she was just made up by Plato, but there seems to be no evidence at all that this is the case; it is true that this is the only source mentioning her at all, but this is not surprising. If one takes her to be a historical personage one is not, of course, committed to saying that everything Plato puts in her mouth is historical as well.
Alcibiades of Scambonidae
Alcibiades needs no introduction, but two things are worth noting that help to indicate a pattern in the dialogue: the dialogue occurs dramatically at the height of his reputation in Athens, not long before the incident of the desecration of the herms; and he was also at the party of sophists in Protagoras.
There are, in addition, a number of slaves, revelers, and participants who are given no names.
We begin in the middle of a discussion with Apollodorus saying that yes, he can answer the question. He remarks that the other day, he was walking to Athens when he was hailed by a Glaucon, who said that he heard a story from Phoenix about a gathering at Agathon's; but Phoenix only had a garbled version, and said Glaucon should talk to Apollodorus if he wanted to know more. Glaucon asks if Apollodorus was at the gathering in question, and Apollodorus remarks that Phoenix must have really garbled the story, because it happened years and years ago, and that he wasn't there, but only (like Phoenix) heard the story from Aristodemus, although he had checked the details with Socrates. Apollodorus then tells his interlocutor, who is never named, that he will tell the same story that he told Galucon.
Aristodemus told Apollodorus that one day he ran into Socrates, who had bathed and put on sandals, both of which were unusual events. Aristodemus remarked on it, and Socrates replied that he was invited to a dinner party at Agathon's, to celebrate the poet's victory in a recent context, so, since Agathon is a handsome man, he was making a special effort to be presentable. He invites Aristodemus along, and Aristodemus accepts the invitation, but remarks that Socrates probably should invent an excuse for bringing him, since Aristodemus is technically uninvited. So they set out together. But Socrates began thinking of something else, and kept lagging, all the while insisting that Aristodemus go on ahead. So Aristodemus arrives at the party.
Agathon welcomes Aristodemus and says that he had tried to invite him, but never could manage to hunt him down, and asks why he didnt bring Socrates? Aristodemus looks around and realizes that Socrates isn't there, and says that Socrates was behind him, but he doesn't know where he is now. Agathon sends a slave out to find him, and the slave reports back that Socrates is standing in front of the neighbor's house and will not come in. Agathon tells the slave to try again, but Aristodemus insists that Socrates does this occasionally, just standing motionless (Alcibiades will confirm this later in the dialogue). If they just let him be, he'll be along when he's ready.
They start eating, and Socrates comes in when he's halfway done; Agathon calls Socrates over to his own couch, joking that perhaps he can pick up wisdom by osmosis. Socrates sits down next to him and replies that he wished one could, because his own wisdom is of no account, but Agathon has been showing his in front of crowds.
After dinner, they pour a libation to the god, sing a hymn, and went through the other little rituals ancient Greeks did in such cases, and then Pausanias asks what they can do to make sure that they don't drink too much. Most of them were at the victory party the day before and so are not in a state for serious drinking; so they agree to drink less at their drinking party, with Eryximachus, the doctor, arguing that serious inebriation is bad for the health. Eryximachus also goes on to propose a way to pass the time. Phaedrus is always remarking that of all the gods, Eros is shortchanged, because not many hymns are made to him. Thus, Eryximachus suggests, they should, starting with Phaedrus, each give a speech in praise of Eros. Socrates agrees quite enthusiastically:
"No one will vote against that, Eryximachus," said Socrates. "How could I vote 'No,' when the only thing I say I understand is the art of love? Could Agathon and Pausanias? Could Aristophanes, who thinks of nothing but Dionysius and Aphrodite? No one I can see here now could vote against your proposal.
"And though it's not quite fair to those of us who have to speak last, if the first speeches turn out to be good enough and to exhaust our subject, I promise we won't complain. So let Phaedrus begin, with the blessing of Fortune; let's hear his praise of Love" (177d-e)
Then Phaedrus gives his speech. Apollodorus says that after Phaedrus, several people spoke, but Aristodemus couldn't remember their speeches. Pausanias goes next. After Pausanias, Aristophanes is supposed to go, but he has the hiccups. Eryximachus recommends a series of hiccup cures -- hold his breath as long as possible, or, if that doesn't work, gargle thoroughly, or, if that doesn't work, make himself sneeze by tickling his nose with a feather -- and offers to go in his place. We are, of course, to imagine that the entire time Eryximachus is giving his speech, Aristophanes is right next to him holding his breath until he turns blue, then gargling and gargling, with hiccups the entire time, and finally by the end of the speech is tickling his nose with a feather to make himself sneeze. Aristophanes has a brilliant speech next. Agathon is next. After Agathon goes, Socrates questions him until Agathon admits that he had no idea what he was talking about when he was praising Eros. Then Socrates begins his speech, which has a speech within the speech, since it is a narrative of his interaction with a priestess, Diotima of Mantinea, in which she teaches him the true nature of Eros. Everyone applauds the speech. In the course of Socrates' speech, Diotima happened to criticize Aristophanes' speech, so it's unsurprising that Aristophanes tries (unsuccessfully) to raise his voice over the noise of the applause to respond to Socrates (perhaps by pointing out that Diotima years before could not possibly have known Aristophanes' story!), and before he can do any respounding, a loud and noisy party of revelers comes crashing into the party, headed by Alcibiades, drunk as a lord and wearing a crown of violets and ivy (212e).
Aristophanes sits on Agathon's couch in order to put his own crown on him, but then discovers (he is, after all, drunk) that Socrates is there, too. Alcibiades insists that they all should drink until they are drunk, or, at least, until everyone except Socrates is drunk, since Socrates can drink without ever getting drunk. Eryximachus lets him in on what they have been doing, and insists that he give his own encomium of Eros. Alcibiades agrees but insists on giving an encomium of Socrates. After the speech, Socrates jokes that Alcibiades is just jealous of the relationship between Socrates and Agathon, and insists that Alcibiades let him praise Agathon. Agathon moves over to hear what Socrates has to say, but at this point another large crowd of drunken revelers comes crashing in, and a number of people, including Eryximachus and Phaedrus, excuse themselves. Aristodemus fell asleep, and only woke up at dawn. Then he saw that Agathon, Aristophanes, and Socrates are the only ones who are still awake and talking, although Agathon and Aristophanes are nodding off. Aristodemus couldn't remember the details of their conversation,b ut he did remember that Socrates was trying to convince Agathon (the tragic poet) and Aristophanes (the comic poet) that the skillful tragic poet should also be a comic poet. When they other two fall asleep in the middle of his argument, Socrates leaves, Aristodemus trailing behind; and Socrates goes straight to the Lyceum to wash up, then spent the day exactly as he always did.
Given the number of speeches, there is obviously a great deal in this dialogue. Some people suggest that there is a sort of progression in the speeches, although I haven't come across any suggestions that are strikingly plausible. It is nonetheless true that each speech contributes something, and that they provide mutual corrections for each other. Phaedrus introduces the idea that Eros is powerful for gaining virtue and happiness (180b) and that Eros is high and good; Pausanias distinguishes a higher Eros, concerned with the soul, from a vulgar Eros, concerned with the body (183e); Eryximachus links Eros with health, harmony, and order; Aristophanes introduces the idea that Eros is "our pursuit of wholeness" and "our desire to be complete" (192e); Agathon introduces the idea that we need to determine first what the qualities of Eros are if we are to talk properly of what Eros does in human beings (194e-195a). Each one (especially that of Aristophanes) could be studied in detail; in particular, looking at the positive contributions and the ironies (e.g., the fact that Pausanias' speech is clearly self-interested, since he is implicitly praising the relationship between himself and Agathon). But as these are in a sense setting up for the speeches of Socrates and Alcibiades, we can move on directly to those.
Socrates attributes to the priestess Diotima his own view of Eros; deliberately ironic, perhaps, given that the role of women in Eros has been pretty limited up to this point -- Phaedrus mentions how Alcestis is rewarded for love even though she was a woman (179b) and Aristophanes insists that his account of Eros applies to men and women alike (193c), but neither Eryximachus nor Agathon bring women in at all, and Pausanias explicitly consigns women entirely to the common or vulgar Eros (181b) and even then only talks about the desire of men for women. According to Diotima, all the other speakers are wrong in thinking Eros to be a god. He is not, because if Eros were a god he would have everything he needs in himself. Eros is instead a daemon, and intermediary spirit between the gods and the human race, partaking something of the nature of both. Likewise, all the previous speakers are wrong in thinking that Eros is beautiful and good. He is not ugly and bad, but a third kind of thing: a drive to the beautiful and good, and which is not ugly and bad, but also by its very nature cannot be regarded as fully in possession of beauty and goodness (otherwise there would be no need for a drive toward, or pursuit of, them). Eros is, so to speak, a direction or orientation to what is beautiful and good rather than that beautiful and good that is its object. This makes Eros a lot like a philosopher, in fact, and this is exactly what Diotima argues: Eros is a philosopher (204b), because philosophers are not fools but they are not properly in possession of wisdom either, being seekers and pursuers of it.
Now, everyone, in fact, has this drive to the good and beautiful, this desire for having the good forever, but we can discern different kinds of ways people act on this. A particular version of this drive is especially suitably regarded as Eros: that which involves "giving birth in beauty, whether in body or in soul" (206b). It is certainly not insignificant that it is only Socrates up to this point who has indicated that Eros in and of itself has any connection with fruitfulness; indeed, he says (what it is difficult even to imagine the other speakers saying) that it is because of its connection with giving birth that the union of man and woman in Eros is a divine affair (206c). Giving birth is the immortality of a mortal creature, a divine thing found in harmony. Thus Eros is in a sense not even love of beauty as such; it is a drive toward birth in beauty (tokos in kaloi). It is only through giving birth, through fruitfulness, that any mortal thing can be said in any way to have the good forever; thus it is this that Eros seeks.
But fruitfulness can be in soul as well as body. Our souls have a desire, an Eros, to give birth in beauty just as much as our bodies do; they give birth to logoi about virtue and virtues themselves. Thus that person who has the version of Eros that is particularly marked out as suitable for the name will begin early with Eros for beautiful bodies, first starting with the beauty of one body, and then recognizing that the beauty of all bodies is the same (210b). Then the lover must rise above this, recognizing that beauty of soul is far better than beauty of body, first seeing the beauty of works and laws, then the beauty of knowledge, and through this, coming to gaze upon "the great sea of beauty" (210d). In doing so, he will come through unselfish philosophy to give birth to reasonings and, ultimately, to knowledge. And only when someone looks at, to have knowledge of, the beautiful itself can one give birth to true virtue, and attain a kind of divine immortality.
This would be a resounding note on which to end, but, of course, we do not end on it. Alcibiades gives his speech, and Alcibiades insists that he will only give a praise of Socrates. Alcibiades says that Socrates is like a satyr, like Silenus (teacher of the wine-god Dionysus) or Marsyas (according to myth one of the greatest musicians of all time). Like Silenus statues, Socrates is ugly on the outside, but inside has little figurines; and Socrates's figurines are so beautiful as to be divine (217a). Alcibiades relates how, after having seen some of Socrates' inner beauty, he pursued Socrates, only to be put off by him, and gives accounts of Socrates at the battles of Potidaea and Delium. He says that Socrates' arguments ar elike this, too:
Come to think of it, I should have mentioned this much earlier: even his ideas and arguments are just like those hollow statues of Silenus. If you were to listen to his arguments, at first they'd strike you as totally ridiculous; they're clothed in words as coarse as the hides worn by the most vulgar satyrs. He's always going on about pack asses, or blacksmiths, or cobblers, or tanners; he's always making the same tired old points in the same tired old words. If you are foolish, or simply unfamiliar with him, you'd find it impossible not to laugh at his arguments. But if you see them when they open up like the statues, if you go behind their surface, you'll realize that no other arguments make any sense. They're truly worth of a god, bursting with figures of virtue inside. They're of great -- no, of the greatest -- importance for anyone who wants to become a truly good man. (221d-222a)
Thus a complete account of Eros requires an account of Socrates himself; the one thing, perhaps, that Socrates himself cannot spend much of his speech providing. But drunken Alcibiades, playing Dionysus to Socrates' Silenus, can complete the picture: now, perhaps, we know what it means to say that Eros is a philosopher. But we readers can notice that Alcibiades in fact never does what would be required (indeed, what Socrates almost outright told him would be required) to win the love of Socrates: he would need to transform himself so that he too was beautiful with virtue inside.
Quotations are from Plato, Symposium, Alexander Nehamas and Paul Woodruff, translators, Hackett (Indianapolis: 1989).