To "not mind" something means to be OK with it, or to not be bothered by it:
A: Sorry it's taking so long.
B: No problem. I don't mind waiting.
You use the phrase "I wouldn't mind ___ing" when you want to do something, but you don't want to seem too eager.
A: So... what should we do now?
B: I wouldn't mind getting a cup of coffee.
In the example above, even if the speaker is really interested in this girl and excited about seeing her again, he probably wants to act cool in front of his friend.
The meaning of "I wouldn't mind ___" can change depending on which word that you stress, though. If you don't really want to do something, you can stress the word "mind":
I wouldn't mind going..."
This means that you don't really want to go, but you will go if necessary.