Note that a pun is a play on words. When you say something that could have two or more potential meanings. Often with one meaning being serious, and the other being humorious. One says 'no pun intended' to either point out an obvious pun that is unintentional and you wish to avoid confusion, or a none obvious pun that is also unintentional, but you want to showcase your mental prowess for having recognized it. Because of the later, it is often looked down upon when said, and has now become something of a joke in and of itself when said as if to make fun of a person who would say it. (Worse yet is when somebody declares 'pun intended'. Puns are considered fairly low humor by most, though I personally enjoy a good pun.)
It means that the person has used some words which produce some kind of word joke, but they didn't mean to make a joke. Usually a pun is a kind of play on words. So sometimes people mean to make a pun, but other times they say something and they don't mean to make a word joke, so they say so, to make it clear. For example the words son/sun are pronounced the same so we can make a joke using those. There are many other words in English which can be used to make a pun or word joke. For example if someone makes a pun I could answer "Very punny" which would be making a joke with the word "pun" that sounds like "fun". Instead of saying "funny" I say "punny".
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
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