Before selling or giving away my computer, is it possible to erase all the sensitive data?

Yes, you can and should erase your personal data before disposing of a computer.  Imagine selling your PC and letting the buyer have access to your personal data, especially if you used the PC professionally.  To delete your personal information, the first and simplest solution is to reformat your hard disk.  You can also buy specialized off-the-shelf software capable of erasing all your personal data and ensuring that it is undetectable and unrecoverable.