Skype is a video and voice-over-IP (VoIP) provider. After signing up for an account, users can call other Skype users and talk with them for free using their computers. If a webcam is available, video chat with up to 3 others is also possible. The Skype interface also has a built-in instant messaging program.
Skype is also able to call mobile and landline phones for $0.026 a minute. Lower rates are possible if you pay monthly. For example, unlimited calling in North America is currently $7.99 a month, unlimited to the world, $13.99 a month.
Some Internet connected TVs (currently Panasonic and Samsung) have Skype built in, though an external camera is required. Some smartphones are able to run the Skype App, perfect for those on unlimited data plans, but limited voice minutes.
Skype claims that during peak hours there are over 23 million users, and currently has over 521 million accounts.
Skype is similar to free services in Gmail, such as voice and video chat. Other similar products include Apple's iChat and Ventrilo.
Wikipedia has articles detailing how VoIP works, the background on Skype, and a list and comparison of all the current VoIP services.