Virus:
A computer virus is a small program written to alter the way a computer operates, without the permission or knowledge of the user.

Trojan:
Trojan Horses are impostors--files that claim to be something desirable but, in fact, are malicious. A very important distinction between Trojan horse programs and true viruses is that they do not replicate themselves.

Worm:
Worms are programs that replicate themselves from system to system without the use of a host file. This is in contrast to viruses, which requires the spreading of an infected host file. Although worms generally exist inside of other files, often Word or Excel documents, there is a difference between how worms and viruses use the host file.

Source:http://service1.symantec.com/SUPPORT...99041209131106

Adware:
Adware is any software application in which advertising banners are displayed while the program is running. The authors of these applications include additional code that delivers the ads, which can be viewed through pop-up windows or through a bar that appears on a computer screen. The justification for adware is that it helps recover programming development cost and helps to hold down the cost for the user. Adware has been criticized for occasionally including code that tracks a user's personal information and passes it on to third parties, without the user's authorization or knowledge.

Source:http://www.spywaredata.com/spyware/s...efinitions.php

Generally,most AVs and programs like Adaware dont remove trojans,they arent meant to do so you see..an AV usually can detect and remove the more popular trojans but they're meant to remove viruses..Adware's meant to remove different types of malware..it's recommended to use applications that are specifically meant to remove trojans for better results.Some programs you might want to look into if you want to make sure you're still infected:
1. TDS-3 from DiamondCS (http://tds.diamondcs.com.au/index.php?page=download)
2. TheCleaner (www.moosoft.com)

These programs generally get rid of most trojans just fine,however..what do you do if a friend managed to code a trojan of his own and kept it just for you?Use a firewall to see what programs have access to the internet from your computer.A few recommended firewalls would be:
1. Sygate (www.sygate.com)
2. Zonealarm (www.zonelabs.com)
3. Kerio (www.kerio.com)
There're several good threads on this site related to firewalls you could pick,try searching for them.