From LQWiki

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 will not respond to user commands; it has often become orphaned when it's parent process was killed, and may not even respond to the lower-powered signals (such as SIGHUP and SIGKILL).