Can spyware actually take over your computer and access other Web sites?
I think you may have been given the wrong term here.
There are a number of examples of malware (including the Subseven Trojan) that can take over your computer (pretty much completely).
Spyware does not attempt to take over your computer, but does try to do certain things you probably won't like. Some will send a Web site a report on all the sites you visit. Others will send other bits of information about your computing habits.
Quite often, they will also access Web sites to download ads to be displayed inside the program which brought the spyware to your computer in the first place.
All in all, most users prefer to get rid of spyware. The reason why it earned this name is that it was "spying" on you and then sending reports back to someone without informing you of this activity.
For more information on this topic, visit these other SearchSecurity resources:
Ask the Expert: Removing spyware
WhatIs Definition: spyware
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