My Apple ID password isn't being accepted even though I know it's correct. What can I do?

This problem has been going on for a while, I can't remember how long exactly but months at least. I've tried changing my password but my computer (Late 2009 Macbook) won't accept it. I can still log in online using my password but every time I enter it into my computer to try to install a new program it says it's incorrect.

I know my computer isn't that old but I needed to take it into a computer repair shop a while ago to have the hard drive (I think, could be different) replaced. Could this be a problem with my computer malfunctioning? Should I take it into the repair shop again?

Less than two hours ago I reset my password but the computer won't accept it. However, I was still able to log into this forum with it. What can I do to fix this problem?

Thanks