Yeah, there's lots of good info on that site. It should help. I can start you off though.
Adware just delivers ads to you.
Spyware logs what you do online and sends the info home.
The line between spyware and what's commonly called a trojan gets fuzzy, because spyware meets the definition of a trojan. It doesn't self-replicate, and it downloads secretively or by deceptive means. The main difference between what's generally called spyware, and what's generally called a trojan is Spyware seems to have a commercial motivation.
A spyware trojan might do something like watch where you're going, and deliver ads to you based on what you watch. It would download those ads from a remote server. It might also change your homepage, and change your default search engine.
Mantice is something different. It's a backdoor trojan.
Trojans are programs (often malicious) that install themselves or run silently on a victims machine. They do not install or run automatically, but may entice users into installing or executing by masquerading as another program altogether (such as a game or a patch) or they may be packaged with hacked legitimate programs that install the trojan when the host program is executed.
Generally you click on something to get that kind of malware. For example you might get a message from someone on an instant messenger that says "click this to see some really cool pictures".
You're right by the way. It's possible to get malware by viewing an evil website. It's usually delivered by an Active X control if you use an insecure Internet explorer browser. Even previously secure browsers such as Firefox are now vulnerable if you click for a plug-in. This is usually commercial spyware that's delivered in this fashion. There are legal issues that commercial spyware has found a way around. Commercial spyware is legal, because they claim they warn you (sometimes it's a lie, but prove it).
New vulnerabilities appear to be popping up primarily affecting newer operating systems like XP that allow users to be exploited remotely. I was reading about one that can get you if you view WMF files. Then there's the sasser worm.
There's a scary thing about Mantice. I was just looking at a graphic of it's controller, and it appears to have a keylogger function. This means it can watch what keys you push and send the info home. Mantice also appears to able to take control of some of the functions of your machine.
Mantice Controller
It's good to have a firewall that blocks outgoing messages; something like Zone Alarm. This way if anything unauthorized tries to call out, you can stop it. Some malware can shut your firewall down, but from what I can find on Mantice, it doesn't appear to do that.