A seemingly useful program you install on your computer, not knowing that it contains another (hidden) one that looks for and sends passwords and other data, or may start a DoS (denial of service) attack. A keylogger, for instance, would record all keys you press, and send this from time to time to the hacker. Of course these reports would also contain passwords. Others open ports---so called backdoors---for a hacker attack. Remote access trojans (RATs) can do all kinds of things on your computer, examples are NetBus, Back Orifice, or SubSeven.
Trojan horses don't reproduce themselves, but are often contained in worms.
Spyware is a special form of trojan horses, sending information about your data or your net activities to somebody. Very often you accept it, somewhere hidden in the licence there is a mentioning of the spyware function. There are special spyware removal programs, even freeware again, like "Ad-aware" or ""Spybot".
Dialer may also be considered as trojan horses, although again you usually agree to it, hidden in the licence. A dialer independently dials expensive numbers. Of course this only works with dial-up connections or ISDN, not with DSL.
Erich Prisner, December 2004