In addition to Pompeiian graffiti and other epigraphy, much of which was quite crude, we also have the books for a number of comedic plays by Plautus and Terence and satirical poems by Horace and Juvenal.

And what were they like? Much like humor in modern periods, you'll find different styles in different authors and at different times. Jokes scrawled on walls tended to be crude insults and/or puns. Plautus and Terence had a number of stock characters that they used to poke fun at social conventions in ways that are reminiscent of modern sketch comedy or slapstick. Horace and Juvenal is much more ironic or sarcastic in most of his satirical poems, pointing out the ridiculousness of politics and everyday life.

I got a book as a gift once, called 'A Greek visits the doctor, humour in antiquity' by Patrick De Rynck and Mark Pieters. It's full of examples of Greek and Roman humour. A bunch of examples isn't really a decent answer to your question, but I figure if someone does give you an answer, they might be interesting as a bonus. I'll try to find English translations of the original texts, but if I can't find any, I should note that this'll be Greek/Latin translated into Dutch translated into English - so not all that literal. Here's a selection:

Mocking poems by Archilochos

As a fig tree on the crag, feeding hordes of crows She's friendly to all comers, Pasiphile the welcomer

...

As on a straw a Thracian man or Phrygian sucks his brew, forward she stooped, working away.

Aesop's fables

Hermes wanted to know how much people valued him, so he assumed a human form and went into a sculptor's workshop. He saw there a statue of Zeus and he asked how much it cost. The man said that it cost a drachma. Hermes smiled, and asked how much the statue of Hera would be. The man named a still higher price. When Hermes saw a statue of himself, he expected that he would be reckoned at an even higher price, since he delivered the messages of the gods and brought profit to mankind. But when he asked how much the statue of Hermes would cost, the sculptor replied, 'If you buy those other two, I'll throw this one in for free!'

Laconic sayings

Plutarch on Sparta:

King Philip had entered Spartan lands and sent them a message: 'What will it be? Should I come as a friend or as an enemy?' The Spartans replied: 'As neither.'

When someone yelled: 'They enemy is near to us,' Leonidas replied: 'Then we are also near to them.'

Cynicism

Antisthenes:

When someone mentioned to him that not both his parents were free citizens, he replied: 'My parents aren't both wrestlers either, but I am.'

Diogenes:

Upon seeing a zither player that kept being abandoned by his audience, Diogenes greeted him with the following words: 'Good day, rooster.' When the man asked Diogenes why he greeted him in this way, he replied: 'Because your singing makes everyone rise.'

Demonax:

He once, for daring to laugh at an athlete who displayed himself in gay clothes because he had won an Olympic victory, received a blow on the head with a stone, which drew blood. The bystanders were all as angry as if they had themselves been the victims, and set up a shout--'The Proconsul! the Proconsul!' 'Thank you, gentlemen,' said Demonax, 'but I should prefer the doctor.'

Obscene poetry by Catullus

What is he doing, Gellius, who has an itch with mother and sister and stays up all night with tunics cast aside? What is he doing, who does not allow his uncle to be a husband? Do you know the weight of crime he undertakes? He undertakes, O Gellius, so much as neither furthest Tethys nor Oceanus, father of nymphs, can cleanse: for there is no crime which can go further, not even if with lowered head he swallowed himself.

Gellius once heard that an uncle is supposed to scold if anyone should say or do any hanky-panky. Lest this should happen to himself, he screwed his uncle's own wife and swore the uncle to silence. Now he does what he wishes: even should he force his own uncle to suck him off, the uncle would say not a word.

Cicero

When during a heated debate, Metellus Nepos kept repeating: 'Who's your father, anyway?' Cicero replied: 'Your mother made that question more difficult for you than for me.'

Philogelos - the friends of laughter

This is the oldest collection of jokes, written by an anonymous Greek writer somewhere in the fourth century AD. You can actually read the entire thing here and listen to Jim Bowen read out some of the jokes.

I've got more, but I'll keep it at that for now.