Nagware is a type of software that includes "nags" -- prompts to the computer user to perform a task or be aware of something. A very common piece of nagware is Microsoft's Windows Genuine Advantage (WGA) that runs on its operating system. If Windows Genuine Advantage is installed from Windows Update and fails to validate an installation of Windows XP, it will turn on Windows Genuine Notifications - this is the actual nagware.
The notifications will basically "nag" the user on boot, on shut down and during a user session that the software failed a validation test and prompts the user to visit a Microsoft link to fix the problem (the fix is generally to buy a discounted copy of the software so that the user has a legitimate license). On Windows Vista, validation software will disable certain functions of Windows Vista, such as Aero themes.
The intent of nagware is to simply annoy the user enough to act. This makes it suitable for shareware programs, like WinRAR, that prompt users who have not registered or obtained a full license for a program by reminding them on launch of the software, or during use of the software. It might also disable some features of the software. In some cases, users are forced to wait until they can use a program while a nagware prompt is displayed.
In the case of Windows Genuine Advantage and others, cracks and patches are very popular because they can stop the nagware from running on Windows start-up and even remove limitations on Windows Update for users running invalidated copies of the Windows operating system.
In any case, it would not be going too far to suggest that nagware is probably a very effective tool at getting users to act, whether it is paying for software or downloading updates or whatever the case may be.