A pun is a joke specifically based on words that sound like other words, or words that can have more than one meaning. I think they exist in most languages. In the US, they are also called 'dad jokes.'
If you google examples of puns, you'll find 1000s.
Two antennas got married. The ceremony was mediocre, but the reception was great.
A bicycle can't stand on it's own because it's too tired. (two-tired).
"No pun intended" just means you said something that might be interpreted as a pun, but you don't mean it to be a pun. Actually, there is even a pun about that:
I sent in ten puns to a pun contest, hoping one would win, but no pun in ten did. (no pun intended)