A Trojan Program, is based on the story about the fall of the city Troy. The walls were too effective for early armies to defeat, and the resources for the encircling army were dwindling fast. The Attacking general however had a plan, he had a large hollow horse statue made, and put it on a cart, loaded a group of infantry into the horse, and hauled it out in front of the main gates to the city, then he retreated, and waited until nightfall.
The Trojan people thought that the sculpture was a gift indicating defeat, and so they hauled it into the city and put it in a park. Later that night, the soldiers climbed out of the wooden horse, and proceeded to open the doors of the city, letting the attacking army in.
A Trojan Horse is a program, that you can't resist letting into your computer. Whereupon, when it is run, it triggers a malware secondary stage, that can use the resources it is given to do something nasty.
With the advent of the Internet, many trojans are downloaded every day by unsuspecting users, loading spyware, adware, and even viruses to their computers without notification. The most destructive ones are those that apply rootkits, and those that convince the computer that it is being attacked constantly by virii.
Think twice before you download that free program that everyone is raving about, you might be loading a trojan.