How are buffer overflows used to exploit computers?

Question

How are buffer overflows used to exploit computers?

How is one able to execute arbitrary code simply by causing stack or heap overflows?

I understand that

Answer 1

This is the most widely known document on the subject: Smashing the Stack for Fun and Profit

However, 'stack overflows' have nothing to do with buffer overflows. Stack overflows are generally just an error case in bad code that can't be exploited for anything outside of a crash (DoS).

Edit: You also asked about heap overflows. This is a good document on the subject: http://www.w00w00.org/files/articles/heaptut.txt