Even though a trojan horse is sometimes also called Trojan, it is more a Greek. The Greeks built the so-called "trojan horse" in the fight about Troja to get into the town, so they are the real snoops ;)

A trojan is a program that has gotten onto your machine without your knowledge and contains malicious code, that would for example allow persons using another computer to connect to yours over a network. Typical trojans are open to anyone trying to connect (any person on your local network or even the internet). Special trojans are designed to make your machine accessible just to the person who infected your computer with the trojan.

The access an outsider can gain using a trojan on your machine can be nearly anything. From watching all your behaviour (like a keylogger), manipulating your computer to basically doing anything you can also do using your keyboard and mouse.

Your computer can get infected with a trojan by multiple ways. A person with physical access to your machine can place it there, but you can also accidentally install it yourself by opening an unknown email attachment that by chance contains a trojan.

According to some definitions, trojans are also programs that sneak into other programs, for example to gain access. As these programs are consciously used by the other, they are not trojans, but are backdoors.