Yesterday, I knocked my laptop off of the desk (it hit the chair and landed on the carpet) and it seemed fine. There was no outward damage; I gave it a little shake to check if anything inside was knocked loose: nothing. It was on when it fell (I've heard that it could damage the system even more when turned on) but it didn't show any signs of disturbance (no shutting down, etc.). It starts up normally and it runs okay; Firefox worked wonderfully, but when I ran a second program with it (SAI Painter), both Firefox and SAI ran extremely slowly and then eventually crashed. I had to restart the laptop and it's working well now, but I'm extremely worried about it. I heard it might be a problem with the hard drive. How would I be able to tell? Is there a possibility that nothing is wrong? Ought I backup my data in case the hard drive crashes? Other than that one occasion, it works as if nothing ever happened.

You should always backup and not wait for a problem to occur, by then it may be too late. You may be lucky and the drive may have survived, but I wouldn't take the chance of loosing the data.

Run CHKDSK /R on the hard drive. It will check the disk for weak/bad sectors. It takes quite a while.