Ad-ware is often bundled in with free-to-download programs. It is seen by the programmers as a way to make money to recoup their costs, and enables them to continue distributing free software.
Once installed adware programs display sponsored advertisements to the software user often in the form of popups or a browser toolbar.
Adware is not always spyware, however the sites you visit in your browser may be tracked and sent back to a central server so more targeted advertising can be displayed to you, thus maximising the distributor's revenue.
If you run anti-virus or anti-spyware programs (as you should) they may flag up some programs guilty of displaying sponsored messages, if so, and you feel your privacy is respected, and you use and are happy with the programs they support, it is likely they are safe to leave on your PC.
If you choose to remove adware elements, certain legitimate programs (i.e those supported by the adware) may not function anymore, so be careful what you remove.