A trojan horse is any code that appears to have some useful purpose but also contains code that has a malicious or harmful purpose imbedded in it. A Trojan often also includes a trapdoor as a means to gain access to a computer system bypassing security controls.

Wikipedia defines it as:

A Trojan horse, or Trojan, in computing is a non-self-replicating type of malware program containing malicious code that, when executed, carries out actions determined by the nature of the Trojan, typically causing loss or theft of data, and possible system harm. The term is derived from the story of the wooden horse used to trick defenders of Troy into taking concealed warriors into their city in ancient Greece, because computer Trojans often employ a form of social engineering, presenting themselves as routine, useful, or interesting in order to persuade victims to install them on their computers.

The following answers are incorrect:

virus. Is incorrect because a Virus is a malicious program and is does not appear to be harmless, it's sole purpose is malicious intent often doing damage to a system. A computer virus is a type of malware that, when executed, replicates by inserting copies of itself (possibly modified) into other computer programs, data files, or the boot sector of the hard drive; when this replication succeeds, the affected areas are then said to be "infected".

worm. Is incorrect because a Worm is similiar to a Virus but does not require user intervention to execute. Rather than doing damage to the system, worms tend to self-propagate and devour the resources of a system. A computer worm is a standalone malware computer program that replicates itself in order to spread to other computers. Often, it uses a computer network to spread itself, relying on security failures on the target computer to access it. Unlike a computer virus, it does not need to attach itself to an existing program. Worms almost always cause at least some harm to the network, even if only by consuming bandwidth, whereas viruses almost always corrupt or modify files on a targeted computer.

trapdoor. Is incorrect because a trapdoor is a means to bypass security by hiding an entry point into a system. Trojan Horses often have a trapdoor imbedded in them.

References:

http://en.wikipedia.org/wiki/Trojan_horse_%28computing%29

and

http://en.wikipedia.org/wiki/Computer_virus

and

http://en.wikipedia.org/wiki/Computer_worm

and

http://en.wikipedia.org/wiki/Backdoor_%28computing%29