Shareware is software that you can try before you buy.
After a trial period, usually a week or two, you have to decide whether to register it or toss it. In some cases, certain features may be disabled until you register, or you may see annoying reminder messages until you register. In other cases, it may be entirely on an honor system. Usually the fee is simply money, but the author may ask for a postcard, a box of chocolate, a donation to a charity, or anything else.
Freeware is software that may be used and distributed freely. Usually the author retains the copyright.
When an author places software in the public domain, that means that he or she has explicitly given up all rights to control its use. This term is often used incorrectly in reference to shareware or freeware.
