Adware, otherwise known as advertising, supported software is just that. When the user is online, they will be presented with a static display box, a banner, or a video pop-up ad, which will create revenue for the advertiser. If the user clicks on the ad, then even more money is made on a per-click basis. Spyware is a whole different animal, in that when the user is online, the spyware will collect information regarding the user, and it will hack your computer without your permission.
Sometimes referred to as malware, this program will infiltrate your information and send it to others, without your approval. This process is completed by what they call cookies. There are programs to remove malware, or sometimes people must recover their computer to dispose of these programs.