A zombie computer is one that has been taken over (by a cracker, using a trojan or virus) and is being used by the cracker to spread spam, trojans or virii.
A zombie process is one that has died, but whose status has not been checked by a wait() call (or similar). Such a process still has a process table entry, because that's where its exit status is kept. It has lost everything its exit status has been requested. It can become an orphan (its parent process may terminate), and like any other orphan process it is immediately adopted by the init process, which is written in such a way that the status will be immediately requested, allowing the zombie to achieve its final rest.