Like its name implies, spyware is a form of malware designed to spy on a system or user activity. An infected machine will usually display no outward signs of harm, and most spyware is designed to be very stealthy and hard to spot.
Sometimes the spyware is relatively benign, in the form of adware which will help deliver specific ads to a user. That is serious enough, but more severe malware could be scanning traffic for information of value, which it can then send back to a server controlled by the hacker. That data could include passwords, financial information, and anything else that is handled by the computer, so spyware can be extremely dangerous.