A botnet (or robot network as the name implies) is essentially a network of slave client computers all connected for the purpose of malicious use (Distributed Denial of Service Attacks, monitoring and stealing network traffic, etc.).
One computer, the master, controls all of the other computers, the slaves, and gives them orders to accomplish a task. These tasks range from constantly sending requests to a server to bring it down, to computers attempting password cracks against various websites.
This is all normally controlled from a secure line to each slave, and the master is typically a server set up specifically for this purpose.
