“No pun intended” means that it may seem a pun to the person who’s at the receiving end, but the speaker doesn’t really mean to offend or make a joke. Some happen entirely by accident, and not intentionally.

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:

Example: a man dares his friend to kiss a rather large woman, to which his friend replies “Fat chance! No pun intended.“

Yes, often “no pun intended” does mean “Ha ha, did you just hear that joke I made accidentally (or at least I’m pretending I made it accidentally).”

Sometimes, however, it can actually mean that you didn’t intend to make a joke.

For example:
A man with a broken hand is trying to carry his groceries, and a woman comes by and offers to help. She says, “Can I give you a hand (meaning “help you”)? No pun intended!”

“No pun intended” is often used to draw attention to the pun (with the hope that your audience will find it humorous). Where a gaff has been made, like the above example with the broken arm, I doubt that the apology for the inappropriate remark would be phrased as “no pun intended,” unless one was reasonably sure that despite being inappropriate the funny side of it would be seen.

It means that someone was trying to make a pun about something.

Sometimes, when a person accidentally makes a pun while talking, they will say “No pun intended”.

Other people (me) purposefully make puns whenever they can, so they say “pun intended”.