Adware is the name for software (normally free to the end-user), for which the author receives payment by presenting the user with a continual stream of advertisements. Most anti-virus protection systems do not detect adware.
Spyware is the name for software containing a trojan component whose purpose is to phone home and send back to the author or some other agency reports on your activity, especially on your download and web-browsing habits. Many downloadable programs and utilities are spyware, particularly distributed file-sharing systems or download agents.
Like a virus, a worm is a trojan with self-replicating ability, but the term worm tends to be used where the malware is an e-mail message which is capable of replicating many similar infectious e-mail messages. This is normally done by sending copies of itself to every entry in your address book: thus the worm tends to spread rapidly amongst friends who work together.
A virus is a trojan with self-replicating ability. In English, the correct plural of virus is viruses. A virus is a piece of computer program (or macro) which attaches itself to another (innocent) program, the carrier. When the carrier is executed, the virus gains control of the computer and copies itself into other programs within the computer, which themselves become carriers. This normally happens within a single PC, which accumulates multiple copies of the virus.
An abbreviation for Trojan Horse, something which looks attractive to the gullible, but containing a secret payload with malevolent intent. The original story of how the Athenians broke into Troy by creating a wooden horse (secretly containing Athenian soldiers) which the Trojans took inside their city walls is told in Book 2 of Vergil's Aeneid. The original Latin contains the famous quotation Timeo Danaos et dona ferentes: I fear Greeks bearing gifts, nowadays jokingly rendered as I fear geeks bearing GIFs. The original wooden horse stratagem remains an accurate analogy of how trojans infect computers today. All computer viruses and worms can be considered to be sub-classes of trojan, while the term trojan tends to be reserved for cases where the malware has no self-replicating ability. Examples of well-known trojans are Back Orifice and SubSeven.