Spyware is software that is used to infiltrate your computer, gathering information and sending it to third parties without your consent. Spyware can also take control of your computer without your knowledge.
Adware is very similar to spyware and is typically installed without your consent. “Adware” stands for advertising-supported software. This kind of software will infect your computer and display unwanted advertisements in order to generate revenue for its author. Some free software may serve a desired primary purpose (a game, a utility), but also install adware to generate revenue --- you may willingly install it for its primary purpose but also get adware installed as a result.
To verify if your computer is infected, run trusted security and antivirus software, and scan for spyware. You can also check your programs list to see if you have any positively identified spyware installed. If you are suspicious about a program, searching online can usually uncover evidence if it contains spyware or adware.