Trojan viruses, named after the Trojan horse that was gifted to the ancient city of Troy by the Greeks. The Greek warriors hid themselves in a giant wooden horse that was left outside the gates of Troy. The people of Troy brought the horse inside the gates, thinking it was a gift. Late that night the warriors broke out of the horse, killed the Trojan guards and opened the gate allowing the rest of the Greek army in to take the city. A Trojan virus works the same way. It is written within a usable application. The user installs the program thinking it will do something useful for their computer, but after it is installed the viral part of the program lets loose with whatever the intended damage is.
Trojan viruses,
A Worm is another common virus, in fact it is probably the closest thing to a real virus. It is self-replicating, meaning that, other than its specific goal of damage to the target system, its other main goal is to copy itself and spread to any systems in direct connection with the original. For instance an email virus that, once it installs in its target and does its damage, finds the target's address book and attempts to copy itself out to all the entries in the book.
Trojan viruses,
These two viruses are the most common,