Shareware is copyrighted software that is distributed by authors online services.  It is commercial software that you are allowed to try out before you pay for it. If you try a Shareware program and continue using it, you are required to register it (or purchase the Licensed version). Shareware is a distribution method, not a type of software.

Freeware is publicly distributed programs that give users a chance to try software and continue to use it for no payment at all or just for some acknowledgement. The authors are still protected by copyright laws, and retain the rights to the programs, but allow people to freely copy and use them.

Public domain is a bit different from the above two. It has a very specific legal meaning.  It means that the creator of a work (in this case, a piece of software) who had legal ownership of that work, has given up ownership and dedicated the work "to the public domain".  Once something is in the public domain, anyone can use it in any way they choose, and the author has no control over the use and cannot demand payment for it.