You can read Laches online in English at the Perseus Project and in French at Wikisource.
(in order of appearance)
Lysimachus son of Aristides
This is the same Lysimachus mentioned in Meno and De Virtute as being an ordinary son of a great man; Lysimachus's father, Aristides, was known as 'the Just'. He is the oldest character in this dialogue, and knew Socrates' father, Sophroniscus.
Nicias son of Niceratus
Nicias was a fantastically wealthy Athenian whose father had made a fortune in using slaves in the silver mines; he was supposedly very generous with his wealth, and this gave him considerable influence. After the death of Pericles he became the main leader of what might be called the 'dove' faction in the Athenian Assembly, and thus the main opponent of Cleon, who was in the opposing 'hawk' faction that wanted to pursue the war further. He was a general several times, and tended to choose those options that would reduce conflict. After the death of Cleon in the Battle of Amphipolis, he (with the help of Laches) was able to negotiate the Peace of Nicias between Athens and Sparta, but at that time Alcibiades and others also began advocating the Syracusan Expedition. He was politically forced into becoming one of the generals of the Syracusan Expedition that he opposed, and indeed, despite his opposition he was soon the only general left to lead it, since Alcibiades was recalled and the third general, Lamachus, was soon killed in battle. He spent a significant portion of the final part of the expedition sick, and then was captured and killed by the Syracusans, despite the attempt of their Spartan allies, who respected him, to prevent it.
Laches son of Melanopus
Laches was also a notable Athenian general and a political ally of Nicias. Both this dialogue and Plato's Symposium indicate that Laches and Socrates fought together at the Battle of Delium, with the latter doing so with exemplary courage and self-possession. Laches died in the Battle of Mantinea.
Aristides son of Lysimachus and Thucydides son of Melesias
Aristides and Thucidydes are present and have (together) one line in the dialogue. Aristides would become a student of Socrates, but Theaetetus says that he left Socrates too soon and fell in with bad company. Thucydides, who is not to be confused with the historian (although he may possibly have been a cousin of some kind), is mentioned in Theages as still a student of Socrates.
Interestingly, Socrates doesn't speak in this dialogue until a fair way into the beginning. Once there he takes the discussion over quickly enough, though.
Melesias son of Thucydides
Melesias's father was one of the major politicians of the early Peloponnesian War. He himself was active in politics, and occasionally shows up in historical records, but as little more than a name. His most significant role was as a member of the oligarchical Four Hundred immediately after the overthrown of the Athenian democratic government.
424 - Battle of Delium
422 - Battle of Amphipolis
421 - Peace of Nicias
418 - Battle of Mantinea
415 - Syracusan Expedition
The Plot and The Thought
Lysimachus opens the discussion by remarking on Stesilaus's demonstration of fighting with armor, which they have just seen. He is particularly interested in the question of whether it is important for boys to learn this, since he and his friend Melesias are considering whether to include it as a major part of their boys' education. To this end, he asks Nicias and Laches, two experienced soldiers and important generals, what their view of the matter is. Nicias agrees to give his opinion, as does Laches, but Laches notes that if Lysimachus is asking for advice, he should also ask Socrates, who is also present, because he comes from the same deme as Lysimachus and has experience with how boys are educated. Nicias also vouches for Laches, noting that Socrates recently recommend a music teacher, Damon, for his son, and it is working out well.
Lysimachus agrees that this sounds good, but suddenly remembers that his boys are always talking about a Socrates. (Lysimachus already knows Socrates as the son of Sophroniscus. But Socrates was a common name in ancient Athens -- indeed, even in Plato's dialogues there is another character named 'Socrates'; Lysimachus just hasn't connected the two.) The boys confirm that it's the same one, and Laches adds that Socrates' career as a soldier has been exemplary, giving his personal testimony of it at the Battle of Delium. Thus it is settled; but Socrates responds that Nicias and Laches are both older and more experienced, so they should go first, and he'll just supplement their advice if anything needs to be added.
Nicias goes first. He thinks it's a good idea for boys to study armor-fighting. It improves their bodies and gives them experience in weapons of war, which could be useful in battle. It can serve as a gateway study to other, more important studies, like tactics. And, in addition, it can make someone braver (andreioteron) and more fierce-looking, which are important on the battlefield.
Laches remarks that the problem with this argument is that it always seems a good idea to learn anything. But if we look at actual experience, we find reason to question the value of this supposed skill. For instance, everyone knows that the Spartans are the best fighters, so if this were a skill of real value to fighting, one would think that the teachers of it would especially attempt to interest them; but, suspiciously, they avoid Sparta completely. Further, Laches claims, he has seen this skill in action, and it is unimpressive. (He has a very funny story about Stesilaus himself and his invention of a 'scythe-spear'.) So either it is not skill at all, or it is a skill of very little value.
Lysimachus remarks that the vote is split one to one, so Socrates is the tie-breaking vote. Socrates, however, expresses surprise that Lysimachus would make this a matter of majority vote, and through a discussion with Melesias argues that decisions need to be made on the basis of knowledge, which means we should find experts rather than decide important things on majority vote. In this case, instead of focusing on the alleged skill of armor-fighting, we should ask what skill we want in the teachers of the boys, and clearly it will have to have something to do with caring for the soul or life. So who is skilled in caring for the soul and has had good teachers.
Laches at this juncture points out that some people become skilled even without teachers; but Socrates replies that you would not accept their word for it unless they could show you a sample of the product of their skill. So this sets up the investigation: if Nicias, Laches, and Socrates are to give good advice, they should either show who the teachers are who are good, have tended the souls of young men, and have taught one of the three, or they should show what people have become good through someone's influence. Socrates remarks that he has not himself had any such teacher; he had no money to pay the sophists who claim to be able to teach in this way, and he was unable to discover it himself. But perhaps Nicias or Laches will fare better, and either be able to identify their own teachers or show that they have learned the skill themselves.
Lysimachus is impressed by this idea, at which Nicias drily remarks that he obviously doesn't know Socrates very well:
You don't appear to me to know that whoever comes into close contact with Socrates and associates with him in conversation must necessarily, even if he began by conversing about something quite different in the first place, keep on being led about by the man's arguments until he submits to answering questions about himself concerning both his present manner of life and the life he has lived hitherto. And when he does submit to his questioning, you don't realize that Socrates will not let him go before he has well and truly tested every last detail. (187e-188a)
But Nicias is willing to go along, and while Laches doesn't like discussions themselves, but he likes it when people harmonize words and deeds, and he knows that he respects Socrates' deeds, having seen them tested in battle.
Socrates then raises the key subject of the dialogue (note that we are already twelve Stephanus pages into the dialogue before we reach its main question): what is manliness/fortitude/courage (andreia)? This is what you're really looking for when it comes to a skill like armor-fighting, so we need to know what it is. Socrates begins by asking Laches, and Laches gives the definition that it is for a man to be "willing to remain at his post and to defend himself against the enemy without running away" (190e).
Socrates, however, raises the point that military strategy sometimes requires defending yourself against the enemy with running away, and that doesn't seem to count against courage. Further, and more importantly, Socrates wants an account of courage that is not confined to the battlefield. Pressed to generalize, Laches suggests that it is a kind of endurance of the soul. Socrates argues that it would have to be a kind of endurance accompanied by prudence (phronesis). But this raises the problem that we in practice sometimes seem to take people to have less fortitude if they have more prudence: someone who endures without knowing that reinforcements are coming might be treated as more courageous or enduring than someone who endures knowing that they are. So Socrates and Laches seem to have come to a puzzle: their deeds don't harmonize with their words. Laches continues to think he knows what fortitude is, but he has at least come to know that he can't put it into words; and he agrees with Socrates that they should endure in the inquiry.
This turns the discussion to Nicias. Nicias, being clever, says he is surprised that Socrates is not using an idea he knows Socrates has used before: that fortitude is a kind of wisdom (sophia). Socrates draws Laches into the discussion, and Laches, true to his promise to endure in the inquiry, asks what sort of wisdom he is talking about. (Note that he is following Socrates' example, perhaps without fully realized it.) Nicias says it is "knowledge [episteme] of the fearful and hopeful in war and in every other situation" (195a). Laches is utterly unimpressed, and after some back and forth on the kind of knowledge, remarks that Nicias thinks that seers are the ones who have courage. Nicias responds by saying that seers do have the appropriate knowledge. He defends himself well against Laches, which Socrates attributes to Damon and Prodicus. Socrates takes over the examination of Nicias, and argues that if fortitude is a kind of knowledge, it could not just be of future things to fear and hope for, but would have to be of goods and evils of any time. But if this is the case, then, fortitude just starts looking like virtue itself.
Laches and Nicias, although not agreeing about fortitude, agree that Lysimachus and Melesias should hand their boys over to Socrates to educate. Lysimachus asks Socrates if he will do it, and Socrates noncommittally replies that they seem to have all three of them shown that they were not appropriate choices, and that what he advises is that they should all try to find the best possible teacher so that they won't remain in ignorance. Lysimachus says that, being the oldest, he is also the most willing to go to school with the boys, and asks Socrates to come to his house early tomorrow to make plans. And Socrates ends by promising.
* Damon was one of the teachers of Pericles; although officially a music teacher, he is said to have advised Pericles on a number of other things, most controversially being the source of the very controversial idea of paying jurors for their service. He is mentioned several times in the Republic, as well as briefly in Alcibiades and Axiochus. Aristophanes mentions him in The Clouds. The man here said to be his teacher, Agathocles, is mentioned in Protagoras as a sophist, and seems to have put himself forward as a music teacher to avoid some of the controversy that had begun to gather around the movement; it is strongly suggested in this dialogue that Damon is doing the same thing, particularly since the sophist Prodicus is also said to be his teacher.
* Each general gives the kind of advice about armor-fighting that one would expect from his reputation -- Nicias gives an answer based on calculation of possible future effects, and Laches gives an answer based on past experience about what a soldier should be like. This is also reflected in their respective definitions of what courage is. As Catherine Zuckert notes in Plato's Philosophers, they also fit the kinds of death each would later die [p. 255]:
As Plato's readers would know, the subsequent fates of the two generals resulted, in part, from the defective understandings of courage they express in this dialogue. Laches lost his life in the battle of Mantinea in 418 because he imprudently gave in to the pleas of his troops to attack rather than wait for reinforcements that would have given him overwhelming superiority. He was ashamed of appearing unwilling to take a risk before his men. Five years later in Sicily, Nicias lost an entire army after he delayed their retreat on account of bad portents. His men were frightened by the portents; he was afraid of returning in defeat to Athens.
Note that this also links directly to Socrates' insistence that matters pertaining to courage should not be decided by vote but by knowledge -- both Laches and Nicias fail to maintain fortitude or courage at the necessary moment because in each case they are more interested in what other people think than they are in what is right.
* Almost all sources that talk about Nicias mention his heavy reliance on signs and omens, so Laches in bringing it up is making a sarcastic reference to his reputation.
* Nicias explicitly notes that he regards Laches and Lamachus as courageous (197c). The three generals on the Syracusan Expedition, which Nicias opposed but went on anyway, were Nicias, Alcibiades -- and Lamachus. This reference to the disastrous Syracusan expedition is certainly not accidental.
* It's interesting that Lysimachus, the oldest person there, is the one who is most willing to entertain the idea that he might need to learn -- neither Nicias nor Laches do. And we know that in other dialogues Socrates will try to encourage others on to the philosophical path by arguing with somebody else entirely -- perhaps Lysimachus was Socrates' target the entire time? And note as well that while Nicias can't get Socrates to take on Niceratus his son as a pupil, Lysimachus is much more successful, precisely because his willingness to 'go to school' gets Socrates to come back.
* William Altman in "Laches before Charmides. Fictive Chronology and Platonic Pedagogy" discusses some of the connections between this dialogue and both Alcibiades Major and Charmides:
In Alcibiades Major, Socrates uses the parallel between the eye and the soul to explain the acquisition of self-knowledge through interaction with another.  In Laches, Socrates explicates an admittedly confusing general statement of methodology (Lach. 189e3-8) with the same analogy between the eye and sight (Lach. 190a1-b2) that he had applied in Alcibiades Major to the soul and the virtue of the soul, i.e. wisdom (Alcibiades Major 133b2-11), and that he now intends to apply first to the soul and virtue generally (Lach. 190b3-c7) and next to the soul and courage specifically (Lach. 190c8-e3). The fact that the identification of human beings with their souls in Laches actually begins with a reference to the eyes (Lach. 185c6; cf. Charm. 156b6) is but one further indication that the entire passage between 185c5 and 190e3 presupposes and is intended to remind the reader of Alcibiades Major. Naturally this kind of claim applies no less to Charmides: in addition to the fact that it revisits the theme of self-knowledge generally, the view of Socrates quoted by Critias (Charm. 161b6) leads eventually to a verbal identity with Alcibiades Major (Charm. 164d4 and Alcibiades Major 131b4; cf. Denyer 2001, 222). But it is not because Charmides is set at approximately the same time period as Alcibiades Major that it echoes the earlier and simpler dialogue: both Charmides and Laches echo Alcibiades Major because they, despite the time-interval between Potidaea and Delium, are best understood as twins.
He also notes that there are linguistic and thematic links between this dialogue and Euthydemus, which is also concerned with the topic of education.
Quotations are from Rosamond Kent Sprague's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 664-686.