In Elizabethan England, jokes were not used in the same way we'd use them now for the most part.

Many of Shakespeare's comedies for instance aren't considered funny (at least not by modern standards), though recently a Prof wrote a paper on this.

It appears that the Elizabethans preferred punnery in their comedy, rather than outright bawdy joketelling to get a laugh. By reverting to the Elizebethan dialect, it became clear that Shakespeare's plays were laced with puns that are simply not clear now because of the change in accents.

For instance, a popular Elizabethan play on words was based around Whore and Hour, which then were pronounced very similarly. The same goes for Loins and Lines.

Source: Prof D Crystal - The Story of English in 100 Words

It's not quite Elizabethan, but there is a joke (which I happen to be amused by) and which appears in at least three joke books between the early 17th and early 18th centuries, beginning with John Taylor's Wit and Mirth (London, 1628), and which might be older. It runs something like this.

A drunken fellow making his way home stumbled into a post. Thinking it a man obstructing him he drew his sword and belaboured it until another fellow came along and said "Hey, do not strike that, it is a post," to which the drunken fellow replied, "If he is the post, why did he not blow his horn?"

In addition to the excellent answers provided elsewhere, as a performer of Shakespeare rather than a scholar there are a couple of others that seem to recur frequently:

• A cuckold (ie a man whose wife is sleeping with someone else) is represented as having horns on his forehead, and is generally a figure of derision.

• Syphilis was the "French Disease", and wordplay involving "French" was often a joke on syphilis and its symptoms (for example, Peter Quince in Midsummer puns on a "French Crown" being both a gold coin and a bald head because syphilis makes your hair fall out).