Latin has a number of puns, and they're all relatively common. They're used by Terence and Plautus quite a lot (and the tradition of using puns in comedy goes back at least to Aristophanes, who's particularly well-known for finding or inventing dozens of words that sound just like women's...naughty bits). They're also a feature of poetry, not for humorous effect (presumably), but to impart double meaning on a line. I was just reading Propertius 3.5 the other day, in which he uses one of Latin's favorite puns, the pun with luxit. Luxit can either be the perfect third person singular of lugeo, to mourn, or of luceo, to shine--in that particular poem, Propertius uses the future perfect form (luxerit) and the line can mean two completely different things, both of which are technically possible. Cicero uses this same pun at least once in the In Pisonem...for some reason this is the pun that stands out in my mind the most (it's really extremely common). Greek likes puns a bit more than Latin (or at least Aristophanes does), but Latin has its fair share of them--do note, though, that not all puns are humorous. The luxit pun doesn't really mean anything funny.

EDIT: Oh gosh, I forgot the most famous funny (and dirty) Latin pun out there! Cicero, in one of his letters to Atticus, complains that he was walking to the forum with his arch-nemesis, Clodius, making smalltalk and pretending not to hate each other, when Clodius decided to prove what a bigshot he was. Clodius, says Cicero, complained that his sister, who was married to the consul of that year, had only given him a couple feet of seats in which to seat his friends and dependents--as a consul's wife she could grant him as much room as she wanted, and was just being stingy. The purpose of the remark was to subtly brag to Cicero of how important even his sister, who was of course not part of politics directly, was. Cicero didn't take the bait, and instead told Clodius that he shouldn't worry about one foot of his sister's, because he could always raise the other. The joke is that "pedem tollere," to raise a foot, is a vulgar Latin idiom for "to have sex with," and Cicero had long accused Clodius of incest. Cicero tells Atticus that he doesn't think it a pun particularly worthy of an ex-consul (as Cicero was), but that it was too good an opportunity to pass up

It's not strictly punning, but Suetonius recounts a lot of sniggering about the sex life of Julius Caesar in his Lives of the Twelve Caesars.

 

An early rumor of an affair with King Nicomedes of Bithynia was often joked about, by Cicero and even by his own troops:

Finally, in his Gallic triumph his soldiers, among the bantering songs which are usually sung by those who followed the chariot, shouted these lines, which became a by-word:

"All the Gauls did Caesar vanquish, Nicomedes vanquished him;

Lo! now Caesar rides in triumph, victor over all the Gauls,

Nicomedes does not triumph, who subdued the conqueror."

 

His prowess with the ladies and affairs with other men's wives was also a source of mirth among the troops:

That he did not refrain from intrigues in the provinces is shown in particular by this couplet, which was also shouted by the soldiers in his Gallic triumph:

"Men of Rome, keep close to your consorts, here's a bald adulterer.

Gold in Gaul you spent in dalliance, which you borrowed here in Rome."

 

Conversely, Caesar was known as a bit of a lightweight when it came to alcohol, leading to "...a saying of Marcus Cato that Caesar was the only man who undertook to overthrow the state when sober."

When it came to gossip and vicious back-biting, the Romans - as in many other spheres - generally excelled.

(Source: Suetonius, The Lives of the Caesars - The Life of Julius Caesar