I've read every definition here and they all get this terms' meaning slightly wrong. Firstly a pun is a
play on words, intended to be semi-absurd and comical by describing a thing or situation using words that could also describe the situation with another meaning. For instance suppose there was a skinny farmer picking berries, but he doesn't find that many, another person might observe "looks like
slim pickins".
Puns often are the worst kind of joke because they do not require much
cleverness.
When someone says no pun intended, however, it is when they say something that could be taken to have two meanings; an obvious one, and a slightly "off" interpretation; and they want you to know they are not making an intentional play on words and want you to take what they said to be the obvious meaning: