These are software programs distributed free, but with embedded advertisements. You will be required to watch the advertisement every time you open the program. Most adware are clearly marked and you know what you are getting when you download and install the application but in other cases the software authors get the software installed with out explicit permission from the end user.
They can also be particularly notorious for being hard to uninstall and usually get installed on your system through deceptive means by being bundled along with other freeware. Adware's evil cousins are spyware (see spyware).