In the simplest sense, a trojan horse is any program
that is DISGUISED. It is installed, or run because
the user is not aware of what is inside,
like the story of the Trojan war.
A trojan can be as simple as displaying an unexpected
message as a joke.
It could damage data, format hard drive etc,
depending on the creativity of the writer.
The most common type we usually talk about
is the server that will give someone access to
the victim's computer.
Contrast this with a VIRUS, which is a program
that can make copies of itself, therefore spreading,
essentially having a life of its own.