Botnets are collections of computers that have software installed on them with which the bot master can remotely control them, usually without the consent or even knowledge of the "zombie" computers' owners. The software is often installed via Trojans, worms or other malware downloaded from a malicious web site or email attachment. Botnets can contain hundreds or thousands of zombie computers, and can be used to send spam, create Denial of Service (DoS) attacks, collect information via spyware and send it back to the bot master, etc.
Microsoft has been working through its Digital Crimes Unit (DCU) to disrupt botnets and cut the connection between the botnet servers and their zombies. Last year, they took down the Waladec botnet and now they've successfully taken down an even larger one, called Rustock, which had an estimated one million computers under its control. Read more about that effort here: