What is it?
The term "spyware" is in most cases synonomous with the terms "adware" and "scumware." One definition of spyware is "any software which employ's a user's Internet connection in the background without their knowledge or explicit permission." They may be picked up through emails, surfing web sites and downloading software, especially "freeware" and "shareware" offerings.
What can it do to my computer?
Keep in mind that spyware programs are executable programs running in the background. As such, they are capable of doing anything any other program can do that doesn't require user intervention. For instance, many of them silently monitor your web surfing habits and report back to central servers. Others display popup ads on your desktop or in your web pages as banners. The more dangerous types of spyware are actually trojans, which allow others to remotely access infected computers and take control of them. In short, at best spyware is a nuisance, in its worst form it is a terrible security risk.
What can I do? How do I find out if I have spyware installed?
One of the things you can do is download and run a spyware detection and removal tool, such as Ad-aware by LavaSoft. Such programs have become pretty good at locating and removing spyware, adware and scumware. Secondly, be wary of downloading freeware and shareware. Many of those programs are supported in their "free" modes through the use of advertising, which frequently involves some form of spyware.