Trojans are malicious applications that pose themselves as legitimate software in order to trick users to install them. Once on the victim's machine, it may run any number of malicious process to steal vital information or inflict damage to other software.
Trojan or Trojan Horse is a general term that refers to programs that appear desirable, but actually contain something potentially harmful. It gets its name from the Trojan Horse that was an instrument of war used by the Greeks to gain access to the city of Troy. It looked like a gift of a giant wooden horse, but actually concealed soldiers inside.
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The harmful contents could be anything, for example you may download what looks appears to be a free game, but when you run it, it opens up a port on your computer where a hacker can "remote control" your machine. A trojan's may also carry other payloads like a virus or worm, which then spread more damage.