To "give a damn" it means you care about whatever you're talking about. For Example: Person 1 "Do you even care?" Person 2 "I don't give a damn" Person 2's response is a very harsh and slightly rude way of saying that "I don't care" although "I don't give a damn" can be said casually.

The phrase is a bit tricky, as sometimes people will use the phrase sarcastically. They will say "I could give a damn!" which means they really don't care about whatever it is. They *could* conceivably care, but they don't. That way of saying it is a bit rarer, but it sometimes means the same as "I don't give a damn!" Like HLO mentioned, there are many varieties of this phrase, another vulgar variation is "I couldn't give two f*cks." I rather like that one..

-ignore -don't pay attention

"I don't give a damn" is similar to "I don't give a sh*t" I think the latter is more harsh than the former.