While microcontrollers, such as the Arduino, are good at making LEDs flash and speakers beep, they can do a lot of other stuff. One of the more interesting things you can do with them is get them to talk.
When I say “talk” I don’t mean speak, but communicate. For example, they can talk to a computer they are plugged into (via USB). They can talk to other devices on a network (wired or wireless ethernet). They can even talk to each other, using something as simple as a twisted pair copper connection to something more interesting like ZigBee or Bluetooth.
Some sample applications:
Perhaps you have a weather station which collects windspeed and direction. One of the challenges of having an Anemometer is that it must be placed away from buildings (ex. on the roof). So, you’d have to run a lot of wire up there. Instead, you could have a microcontroller connected to a ZigBee module, all of which is powered by a small solar panel. (Most microcontrollers are very low power and can easily run on a battery/solar combo.) The ZigBee has a 100 meter range (indoors) so you can place it quite a ways away from the house.
Or perhaps you want the ultimate child monitor. You place a microcontroller, ZigBee module, and RFID reader combo next to each deadly area in the house (stove, stairs, paint thinner, dirty magazines, etc). Then you embed a RFID in your childs arm. Every time they get near the RFID reader, the microcontroller picks it up, transmits information via the ZigBee, and alerts the parent. (Since ZigBees work as a mesh network, they hop messages to each other, so you don’t need direct contact between child and base. Every module is a base station of sorts.)
You could be an avid bird watcher, with several bird feeders set up around your land. You could put a ZigBee and microcontroller and a sensor (heat, accelerometer, whatever) on each feeder. Then, when a bird approaches, the info is passed over the mesh network alerting you that there are birds in the area. It could also trigger things like cameras, which would record only when birds are in the area. (Similar to the ones that hunters use for deer; except these cameras communicate back to the base station in real-time.) At the base station, a module could be connected to the internet, sending you an e-mail each time a feeder is used.
You get the idea. There are tons of uses for these low power, mostly inexpensive, devices. The problem is finding a place to learn about them, and for that I would suggest a book called “Making Things Talk”.

You can pick up a copy at Amazon for about $20. It focuses on the Arduino, so if you are interested in learning how to do more complicated stuff with that board, this is a book for you. It starts up easy, and then gets into connecting stuff to ethernet, Bluetooth, and Zigbee. It also does a good job in covering a number of different sensors (probably the most expensive part of many of these projects).
I’m not a big fan of how the book is laid out (trying to be too cool, and thereby making some sections hard to follow), but the information–and ideas–are gold. It also starts you out with little assumed knowledge of electronics, which is nice for beginners; though the book is not dumbed down, especially in the later chapters.
What really impresses me in general about this stuff is how easy it is to get a microcontroller talking with another one over a wired/wireless connection. The connections are pretty straightforward, and all the radio negotiations are handled by the ZigBee/Bluetooth boards. Just give them power, and connect the transmit/receive pins. That’s about it. This is wonderful, because it allows people to take ideas and make something very complicated without needing a degree in electrical engineering and a million bucks.