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Wikipedia:
A pun (also known as paronomasia) is a play on words that transposes the meanings of words with similar sounds. This is usually for humorous effect, although one well known pun of serious intent is found in the Bible: Matthew 16.18:

"Thou art Peter [Greek Petros], and upon this rock [Greek petra] I will build my church."

(Note that while petra is "rock", the word for "stone" in general is petros.)
The word pun itself is thought to be originally a contraction of the (now archaic) pundigrion. This latter term is thought to have originated from punctilious, which itself derived from the Italian puntiglio (meaning "a fine point"), diminutive of punto, "point", from the Latin punctus, past participle of pungere, "to prick." These etymological sources are reported in the Oxford English Dictionary, which nonetheless labels them "conjecture".

Although there are several varieties of puns, there are two main linguistic methods for creating them:

Homographic, in which where the pun exploits a word with multiple meanings. For example: "Being in politics is just like playing golf: you are trapped in one bad lie after another."
Homophonic, in which the pun exploits two words with similar sounds. For example: "A chicken crossing the road is pure poultry [like poetry] in motion."
The compound pun is one in which multiple puns are colocated for additional and amplified effect.