As often with Plato there’s a layered entry: the party being related is remembered at second hand. This is important as it sets up doubt, makes us wonder what’s really going on here.
The game the men decide to play is to each in turn give a speech in praise of love.
Phaedrus: love is very ancient and important, as it’s the best motive for noble and heroic action. No one runs away in sight of their lover. Achilles and Patroclus, pace Aeschylus, weren’t lovers.
Pausanias: two Aphrodites: one common, one noble. The noble one is the proper εραστης/ερωμένος relationship. No activity is all bad or all good – it’s how it’s done which counts.
Eryximachus: love can be applied to all aspects of life, e.g. medicine, and is what maintains a healthy balance between extremes. It’s a kind of harmony.
Aristophanes: love is the essential task of seeking our original “other half“, or close alternative, and reuniting with that person. Much comedy in the ridiculous way he tries to explain the anatomy of these combined people.
Agathon: important in a panegyrics to state the subject’s properties, and then their effects. Love is kind, peaceful, sensitive, and makes the whole cosmos like this, including inspiring all artistic activity.
Socrates: sarcastically points out that he didn’t realise that the rules were not to tell the truth, but make a fancy speech ascribing to the subject if the panegyric all the good qualities one can think of. He then insists on doing it ‘his way’, and reduces Agathon to blabbering that he didn’t know what he was talking about earlier (as love of something implies that the lover lacks the something; love therefore can’t be itself beautiful and desire the beautiful).
He then relates Diotima’s Socratising of himself, patiently pointing out that what people (and animals) in fact love is immortality. She then goes on to the hard bit, the ‘perfect revelation’, whereby men’s love for beautiful men leads them to all beautiful objects, to beautiful thoughts and sciences, and finally to το καλον, beauty itself, the form of beauty. She ends with fusing these two strands, arguing briefly that if a man loves beauty itself he will be loved by God, and therefore immortal himself.
Then chaos, as drunkards break in, led by Alcibiades. It’s very funny, and eventually Alcibiades gives a speech in praise of Socrates himself, how he has successively tried and failed to seduce him, and how his words, valiant actions in battle, and total self-discipline have affected him.
Then chaos, as more drunks take over, this time ending the civilised speeches. In the morning only Socrates, Aristophanes and Alcibiades are left awake, and at the end of the dialogue only Socrates.

So what to make of it all?

There’s lots of comedy, more than usual for Plato. It comes partly from the banter of the party, but also interestingly from the undermining of the earlier speeches. Socrates deftly takes Agathon apart, and, by extension, the other speakers. And by using a woman, Diotima, as his primary authority on love, he is cancelling Pausanias’ assumption that women are baser, less intelligent. Further undermining can be found in the way Diotima in her speech, and Socrates in Alcibiades’, both knock away the foundations of the symposium itself, the Athenian male pederastically-banterous drinking party. Men-only, pederasty, drinking – all fall foul of Socrates’ arguments.