While Adam B's answer was precise, it is quite obvious that he just copied and pasted it from most-likely Nintendo's site.
Let me put it in a way that the average person would be more likely to understand. Wii MotionPlus is an attachment that plugs directly into the port the Nunchuk would normally fit into (do not worry, for games that use Wii M+ and the Nunchuk, there is a port built into the Wii M+ so that you can still use the Nunchuk). What this little device does, is it provides a whole new level of precision in gaming. Movements will now be able to be recognized in a much more "exact" way. Basically, whatever you're doing with the Wii Remote in your hand will be shown exactly as it is being done on your TV. This is not possible with a stand-alone Wii Remote. Your controller with Wii MotionPlus will now be better recognized as it is being held in space; providing a considerably realistic experience.
Games that are currently out that use Wii MotionPlus are at the moment Grand Slam Tennis and Tiger Woods PGA Tour '10.
Games that are not yet released and are compatible are: Wii Sports Resort, Red Steel 2, Virtua Tennis 2009, Kidz Sports - Crazy Mini Golf, Food Network: Cook or Be Cooked, Academy of Champions, Gladiator A.D., The Grinder, and Span Smasher.
Will it be compatible with old games? No. For example, it cannot be used with Wii Sports. There will be no gameplay changes.