The term "Botnet" is generally used to refer to a group of infected computers (sometimes referred to as Zombie computers) running malicious software in the background.
As the name suggests a Trojan is a malware program that masquerades as a legitimate application or file.
Simply put, keyloggers are malware that record keystrokes. Keyloggers can keep track of web sites visited, and time of your visit, as well as what keys you type.
Rootkits are malware designed to take control of your PC without your permission. A rootkit is intended to seize control of the operating system running on the hardware.
Worms are self-replicating malware that uses a network to send copies of itself to other nodes (computer terminals on the network).