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Spyware is just one form of something generally called malware which is software that does something that you probably don't want - often without your knowledge. The three most common types of malware are spyware, adware and vuruses. Spyware is not generally viral in nature although they sometimes use viral entry routes and exploit security holes in microsoft software. It can be anything from a fairly benign piece of software that simply monitors what you look for on the web and reports it to marketing companies to something that grabs your keystrokes and credit card details and reports that to a hacker. Adware can be software that was legitimately installed with a piece of free software (as "payment") or something that was installed using an exploit (security hole). Either way the end result is the same, popup adverts appearing when you least expect them. Viruses attack your computer and often take it over and use its resources for other means - such as sending email.
You may be offered software to clean up your computer but as a general rule don't install it. There are a number of malware removal utilities that are either malware themselves or have convenient blind spots for particular pieces of malware. If you really want to install a piece of security software that you found yourself please ask your friendly geek first :o). The utilities listed below (if kept up to date) should protect you against the vast majority of problems anyway and negate the need for anything else. Unfortunately you will have to learn how to use them but that should only take a few minutes (I might get round to writing some simple instructions at some point).
Firewalls
Running a firewall is a very important first step to secure your computer but contrary to popular belief they do not protect you from all viruses. A firewall only blocks access to the open ports on your computer. Leaving a port open on your computer is like leaving a window in you house open when you go out. A criminal can just climb in steal things and leave. The firewall is the equivalent of closing and locking all your windows. However, even if you have all your windows closed and locked if you invite a thief into your house - for instance someone pretending to read the electricity meter - they can still steal things. As you can see your firewall has worked. It stopped people climbing in through the windows but it can't stop you inviting people in through the front door. In this analogy the front door is your web browser and email client and inviting people in is visiting web pages and opening attachments on emails. Internet Explorer and Outlook (Express or Full Version) are like having a front door where half the population has exactly the same key as you. Yes the door works and locks but you are relying on everyone else being trust worthy and not just letting themselves in!
Utilities
Stinger
This is a free virus removal tool that will catch the most common viruses (there are thousands of viruses but only a few are popular). If can be downloaded from McAfee.
AdAware
A free malware removal tool. It is very easy to use but I have noticed it doesn't always get everything. I suggest at a minumum you run this tool but if you can run SpyBot as well. Adware can be downloaded from Lavasoft.
SpyBot
Another malware remover. Seems to be more comprehensive than AdAware but is somewhat harder to use (actually its easy to use but has a terrible user interface design). SpyBot can be downloaded from Safer Networking.