Malware is software developed for the purpose of doing harm. Malware includes virus, worm, Trojan horse, spyware, hijacker, and certain type of adware.
Trojan is a program that has been designed to appear innocent but has been intentionally designed to cause some malicious activity or to provide a backdoor to your system.
Virus is a program that when running, has the ability to self-replicate by infecting other programs and files on your computer. These programs can have many effects ranging from wiping your hard drive, displaying a joke in a small box, or doing nothing at all except to replicate itself. These types of infections tend to be localized to your computer and not have the ability to spread to another computer on their own. The word virus has incorrectly become a general term that encompasses trojans, worms, and viruses.