The word Malware is short for malicious software, and is a general term used to describe all of the viruses, worms, spyware, and pretty much anything that is specifically designed to cause harm to your PC or steal your information.
Spyware is any software installed on your PC that collects your information without your knowledge, and sends that information back to the creator so they can use your personal information in some nefarious way. This could include keylogging to learn your passwords, watching your searching habits, changing out your browser home and search pages, adding obnoxious browser toolbars, or just stealing your passwords and credit card numbers.
Trojan horses are applications that look like they are doing something innocuous, but secretly have malicious code that does something else. In many cases, trojans will create a backdoor that allows your PC to be remotely controlled, either directly or as part of a botnet—a network of computers also infected with a trojan or other malicious software. The major difference between a virus and a trojan is that trojans don't replicate themselves—they must be installed by an unwitting user.