ASLR is a technique designed to make various types of buffer overruns more difficult to exploit, by moving segments around a bit. The stack could be shifted a few bytes (or pages), the sections of your program (and even the libraries your code uses) can be loaded at different addresses, etc.
Buffer overflows usually work by tricking the CPU into running code at a certain address (often on the stack). ASLR complicates that by making the address harder to predict, since it can change each and every time the program runs. So often, instead of running arbitrary code, the program will just crash. This is obviously a bad thing, but not as bad as if some random joker were allowed to take control of your server.
A very simple, crude form of ASLR can actually be implemented without any help from the OS, by simply subtracting some small amount from the stack pointer. (It's a little tricky to do in higher-level languages, but somewhat simpler in C -- and downright trivial in ASM.) That'll only protect against overflows that use the stack, though. The OS is more helpful; it can change all sorts of stuff if it feels like. It depends on your OS as to how much it does, though.
