Spyware is a general term used to refer to files and software that have been downloaded to and installed on your computer without your knowledge. Spyware is typically responsible for those annoying pop-up ads (adware) and for distributing identifiable information about you and your computer to any number of clandestine people and computers on the internet. It's also used for harvesting email addresses from your address book, archived email messages, web browser cache, etc. and distributing these addresses to systems across the internet that send out spam (junk email). Your computer can become "infested" with a variety of spyware (or particularly malicious spyware) which can cause your internet connection to be saturated with garbage that the spyware program is sending/receiving in the background, slowing down your computer.
In short, it's bad stuff and you don't want it on your computer!
You may not be aware of it, but you probably already have some spyware on your computer.
Microsoft has a lot of information about this, including references (and links) to software that removes spyware/adware, and information about how to prevent it in the future. See this:
Removal and prevention software:
Spybot - Search & Destroy (Spybot-S&D)
Microsoft Security Essentials
Some anti-virus software also removes/prevents spyware/adware.
PC Magazine published an extensive article about spyware in April 2003 that includes a review of some of the software used to remove it. Although the review is somewhat outdated (software updates have since been released), the information is still quite valuable:
Spyware--It's lurking on your machine