What term is used for a computer that belongs to a group of networked computers, all working mindlessly to serve the person who installed the program on the computers?

Zombie.

Explanation:

A zombie is a computer that is connected to the internet or a network but has been infiltrated by a Trojan horse, computer virus or hacker. This activity is not noticed by the owner or user of the computer.

These zombies usually launch DoS (Denial of Service) attacks and mostly spread email spam. They can be used also to get vital information about an organization or government agencies.

Up-to-date security measures must be installed and carefully considered to prevent these attacks.