You can read Lysis online in English at Perseus Project or in Victor Cousin's French at Wikisource. Gregory Sadler has several useful handouts on the dialogue at Academia.edu.
Socrates narrates this dialogue.
Hippothales son of Hieronymus
Hippothales is unknown outside this dialogue, although it is a common name.
Ctesippus of Paeania
Ctesippus is also found in Euthydemus, where he is closely associated with Clinias the son of Axiochus, and he was with Socrates in his last days (he is present, but does not speak, in Phaedo); he is Menexenus's cousin. We know very little else about him. He seems to have been quite outspoken, since both this dialogue and Euthydemus show him as willing to say things others aren't; Euthydemus characterizes him as well-bred but wild. Curiously, we never learn who his father was; we are merely told he is from the deme (borough) of Paeania.
Lysis son of Democrates
Lysis is one of the main interlocutors, of course, a friend of Menexenus and the person on whom Hippothales has a crush. Most of what we know about Lysis outside this dialogue is from monumental evidence -- archeologists have discovered his gravestone and those of his immediate family. He apparently went on to live a long life.
Menexenus son of Demophon
Ctesippus's cousin and best friend of Lysis. Like Ctesippus, he is mentioned in Phaedo as being with Socrates at the end. He is, of course, the other interlocutor in Menexenus.
There is also an anonymous crowd of other boys.
The Plot and The Thought
Socrates narrates that he was going straightway from the Academy to the Lyceum -- two important common areas in Athens -- when at "the little gate that leads to the spring of Panops" (203a) he comes across Hippothales, Ctessipus, and some other youths. (Panops is another name for Hermes.) Hippothales hales him, and persuades Socrates to turn aside to "a sort of enclosure and a door standing open" (203b). It is a palaestra or wrestling school, where the boys hang out and mostly spend their time in discussions. Socrates wants to know who the most handsome boy in the school is, and when he presses Hippothales in particular to say who he thinks the most handsome, the youth blushes. Socrates notes that Hippothales is in love, remarking that however he might be useless in other areas, he has a gift for recognizing lovers and beloveds.
Hippothales blushes even more, and Ctesippus, apparently amused, lets the story out, complaining that Hippothales never shuts up about Lysis, especially when he's drunk, and worse than that are the poems and eulogies, and worst of all is when he gets it into his head to sing about the boy. Socrates demands that Hippothales provide a demonstration of what Ctesippus is talking about, which the boy, of course, refuses to do, denying that he writes love poems and eulogies. But Socrates replies that he does not want to hear Hippothales reciting poetry, but simply what sort of thing he is saying about Lysis. Hippothales remarks sarcastically that Ctesippus can probably say, and Ctesippus doesn't hesitate to oblige, describing (in perhaps exaggerated ways) how Hippothales extols Lysis' ancestors, and things like that.
Socrates chides Hippothales for singing his own triumph-songs -- if he gets Lysis, then everything he's saying will redound to himself, whereas if he fails, he will look ridiculous. It also swells the heads of the subject and makes them harder to catch. Hippothales asks for further advice, and Socrates remarks that he might be able to help if he can talk to Lysis, since he could give an example of the real way one should talk with the youth.
Ctesippus leads him inside, where the boys are celebrating the Hermaea, Hermes being the patron of wrestling schools. There they find the boys playing knucklebones in their best clothes, having finished the sacrifices for the day a little earlier. There they see Lysis with a garland, but at first he's too shy to approach. He only comes over when Menexenus sees Ctesippus and joins them. Socrates starts talking about their friendship. Menexenus is called away, so Socrates begins interrogating Lysis, putting a puzzle to him.
(1) Lysis's parents love him. If so, they obviously want him to be happy. But if happiness is being able to do what you like to do, we get the reverse: they prevent him from doing all sorts of things he'd like to do. So they must not love him. Indeed, they let the house slaves do things that they would prevent Lysis from doing, so in that sense they love their slaves more than Lysis. Lysis suggests that this is because he is not yet of age, but Socrates points out that this doesn't really explain the difference between what they will and won't let him do. Socrates argues that the key difference in such cases is understanding (phronesis): people trust those who understand. If he becomes wise, everyone will be his friend.
Lysis at this point concedes that perhaps he's not very impressive, at which point Socrates looks for Hippothales, who is hovering at the edge of the crowd trying not to look interested in Lysis. He almost says something to him about how this is the way to treat favorites, but refrains from doing so in order to avoid embarrassing Hippothales. Menexenus comes back at this point. Lysis playfully asks Socrates to tell Menexenus what he just said. (Menexenus is a good debater, so one gets the impression that Lysis has a lost a few arguments with Menexenus and wants Menexenus to know what it's like! And indeed, he will say later that he wants Socrates to trounce Menexenus.) Socrates insists that Lysis himself do it, and Lysis promises that he will, but that in the meantime Socrates should tell Menexenus something else.
(2) On the one hand friendship seems to be mutual. On the other hand, it is possible to love without return, and it seems in that case that the lover is a friend who has no friend. Yet again, though, if one loves one's child, the child seems to be the parent's friend, even if the child, after being punished, currently hates the parent. But this is generalizable -- you might love someone who hates you, and thus, it seems, be a friend (philos) to your enemy (echthros, a word, incidentally, that might be familiar to readers of Madeleine L'Engle). Yet it seems absurd to say that you can be friends to your enemies and enemies to your friends. So perhaps there is something wrong with how they are approaching the matter.
Lysis at this point says he thinks they are, so Socrates switches back to him.
(3) It seems that in friendship like attracts like; people say this, anyway. But unjust people don't seem to be drawn together at friends, so this doesn't seem true. So Socrates suggests that what that really means may be that the good are friends with the good and the wicked being divided in themselves, aren't even friends with themselves. But on the other hand, if like is friend to like, then what's the use of friendship -- what can a friend provide you that you can't provide yourself? But yet again, people also say that opposites attract. But then, the hater is the opposite to the friendly, which seems therefore to imply that the hater is friend to the friendly. Thus we seem to be driven to the conclusion that what is neither good nor bad is friendly to the good -- that is, what is neither good nor bad is not the enemy of good, but is unlike (and therefore can need) the good. Socrates links this with philosophy:
And consequently we may say that those who are already wise no longer love wisdom, whether they be gods or men; nor again can those be lovers of wisdom who are in such ignorance as to be bad: for we know that a bad and stupid man is no lover of wisdom. And now there remain those who, while possessing this bad thing, ignorance, are not yet made ignorant or stupid, but are still aware of not knowing the things they do not know. It follows, then, that those who are as yet neither good nor bad are lovers of wisdom, while all who are bad, and all the good, are not: for, as we found in our previous discussion, neither is opposite friend to opposite, nor like to like. (218a-b)
Everyone is satisfied with this, but suddenly Socrates shouts out that they've all put their trust in untrustworthy arguments.
(4) It seems that we are friends for the sake of something -- at least, it would be odd to say that we were friends for no reason at all. But what kind of thing is it for the sake of which we are friends? It's the kind of thing we love -- something dear, something favored, something 'friendly' (in the sense that even we today talk about environments being 'friendly' or computers being 'user-friendly'). But it seems that we cannot have an infinite regress in things that are dear or favored ('friends' in a broad sense, what we love or cherish). So there must be a First Friendly Thing (proton philon), for the sake of which we hold dear all the other things we hold dear. But if this is right, then why would we not simply say that there is only one friend, the first, and that everything else is just a means to it. If this is the case, then the obvious candidate for the first friend is good itself. But we have already suggested that we love that for which we have some need, and thus lack. But if we love good because of our need or lack for it, we love it because of what is not good, and thus for what is bad. Thus it seems that what is friendly is friendly for the sake of what is inimical to us. Yet it seems that if you abolished everything inimical some things would still be friendly. Thus friendship appears not to be based on needing but on having: things are friends because they have a bond that makes it so that they belong to each other.
Menexenus agrees that this seems likely, but Lysis is silent. Hippothales, however, is simply beaming. Socrates then reviews the argument, and is about to draw in some of the older boys there, but the tutors of Menexenus and Lysis show up, dragging along their brothers, and the tutors are loud and drunk and insistent on the boys coming with them. Although all the boys there try to get them to go away, it is futile, so the band starts to break up. Before they do, though, Socrates ends the dialogue with a comment on the whole discussion:
Today, Lysis and Menexenus, we have made ourselves ridiculous—I, an old man, as well as you. For these others will go away and tell how we believe we are friends of one another—for I count myself in with you—but what a “friend” is, we have not yet succeeded in discovering. (223b)
* Despite the extraordinary suggestiveness of Socrates going from Academy to Lyceum, which are best known today for being the locations of Plato's school and Aristotle's school respectively, both locations of course long pre-existed either school. The grove of Akademos was to the northwest of the main city; the Lyceum was just inside, or just outside, the main city on the eastern side, not far from the Acropolis. There seems to be no consensus about what to make of Socrates' claim that he was going straight from Academy to Lyceum by means of the road outside the town wall, which seems to be a long way around. (You can see both on the map below; you can click through to the Wikimedia Commons page.)
* Unlike most aporetic dialogues, where there is an ambiguity about how much Socrates is just letting the argument follow a natural course and how much he is deliberately maneuvering to show the interlocutor's ignorance, in this dialogue Socrates is quite clearly maneuvering; this is set up explicitly by the fact that Socrates is demonstrating the opposite of Hippothales's approach (which gives the beloved a swelled head), and is confirmed by the fact that Socrates stops just short of highlighting this explicitly to Hippothales, after he gets Lysias to admit his ignorance. He is deliberately flummoxing the argument.
* The word for belonging at the end of the dialogue is oikeion, which could also be understood as 'belonging with' -- or, perhaps even better, 'at home with', since it is related to the Greek word for a household.
* Notice Socrates' slyness here, which is also shown elsewhere in other dialogues (e.g., Rival Lovers): he is drawing boys in not by arguing directly with them, but by arguing with someone else. He draws Hippothales toward philosophy by arguing with Lysias; he draws Lysias into philosophy by arguing with Menexenus. He draws all the rest of the boys, who at the end want the discussion to continue, by arguing with both Menexenus and Lysis.
Quotations from W. R. M. Lamb's translation at the Perseus Project.