Arduino boards allow you to learn about electronics or to create new electronic "stuff". You can use an Arduino board to prototype something that you'd like to have manufactured as a printed circuit board (PCB), or use the Arduino directly to program sensors, motors, lights or input devices.
The best way to start is with a kit. There are some basics that you need to learn first about how the Arduino board and the programming interface works. Once you have the basics down, the sky's the limit!
Make Magazine has a whole section devoted to Arduino programming and project ideas.
There are tons of Arduino projects on Instructables as well. There's also a collection of the Top 40 Arduino projects on the web.