Shareware is software that you can try out before you buy. Shareware is copyrighted, but you can download it freely, try it out, and even distribute it around to people you know. However, after using the software for a trial period (70 days for my programs), you must register and pay for it, or delete it from your computer.
Shareware is not really a type of software, it is better described as a method of distribution. By allowing you to try a program to see if it is suitable to your needs, shareware offers some distinct advantages over commercially available, shrink-wrapped software sold through other channels. Since you get to try the program at virtually no cost, you can assure yourself that the software is suitable for you before paying for it. You can't do that with other types of software.
Shareware authors are in business, just like the commercial software companies. Unlike those companies, however, shareware authors are willing to let you try their programs before paying for them. It's a great deal for the user and makes possible the creation of many programs which would not otherwise exist.