Viruses, Trojans, etc. are collectively known as Malware.
Viruses: The term comes from the biological term, virus. Like virii in real life, a computer virus infects an organism and then spreads by replicating itself.
Trojans: The term is again borrowed from the concept of the "Trojan Horse" from Greek Mythology. A Trojan is a computer application which looks completely harmless on the outside. It is only after you install it that you realize what it actually is, but the damage is done by then.
The amount and extent of damage caused by any malware depends greatly on its purpose. Some malware were written only for fun and may display a limerick on your screen, but nothing more. Others may harvest your data and send it out so that it can be sold to the highest bidder. Some malware exist only to break into some other program in your system and then download its payload which is what actually performs the malicious activities.
Some malware will do absolutely nothing to your system. Stuxnet is a famous example of this. It was written to target only a certain computer and would remain dormant on all others.
Yes, you can always remove a malware from your computer manually. You only need to replicate the exact steps that the anti-malware applications perform. It's possible, I've done it, but it's not easy. And more often than not you'll leave something behind. You should use the specialized software available instead.
The easiest way to prevent malware from infecting your system is to use Linux. As explained by Miami Tom Windows has an inherently broken security mechanism. Linux on the other hand is very strong. If you must stay on Windows, use a good anti-malware tools with a firewall.