The word Trojan comes from the classic Trojan Horse in Homer's Iliad. In this story the Greek left behind a large wooden horse outside the city of Troy and sailed off. The citizens brought the wooden horse into town. The horse contained Greek warriors, who then jumped out, killed a bunch of people, and opened the city gates, letting in the rest of the Greek army who had been hiding.
A Trojan generally is not by definition a virus and unlike common belief they do not spread to other programs and other computers like a virus either. However they are one of the leading causes of computer breakings. They can also contain a virus within them.
Trojans are generally programs that pose as a legitimate program on your computer and add a subversive functionality to it. That's when it's said a program is Trojaned. For example a Trojaned login program can be written so it accepts certain passwords for any user's account to give the intruder access to your computer. A commonly used program called Toolkits is actually a suite of several Trojans that can be put on a Unix machine by an intruder. It contains a password sniffer, a backdoor program to let the hacker back in to the machine at any time, replacement binaries for common programs and finally a log cleaner to cover its trace.
Trojans can contain a virus, a password grabber or they can be a RAT (Remote Access Trojan) that is designed to allow remote control over your system. Some Trojans contain built in scanners that automatically scan the Network from your computer, looking for another copies of themselves. The most commonly used Trojan is The Sub 7 Trojan. What makes it common and dangerous is the fact that unlike other Trojans that are written once and forgotten, Sub seven's author provides constant improvements and new versions for his Trojan. This Trojan can be really destructive or just annoying.
Remote access Trojans such as Sub 7 consist of two parts. One part that resides on the victim's computer and the Control module that resides on the Intruder's machine. Using the control interface the intruder can take control of your computer, flip your background picture, Speak out of your speakers, reboot your computer or do serious damage to your machine and do pretty much whatever he or she chooses. However these days most Trojans are being used to turn your computer into a Zombie. In other words the intruder will be able to use your computer and a bunch of other infected computers like yours and turn them into an army of zombies to attack a more important target. This way the intruder can hide his or her own trace and use your computer as a front. This means the victim's firewall will show your IP address in the log as the attacker's IP and you may end up with several complaint letters to your ISP and even lose your account as a result.
So just because you think you have nothing important on your computer doesn't mean your computer itself and your bandwidth is not valuable to hackers. In fact they are quite valuable and worth the hacker's time.