It's a standard feature of most every house you've ever seen, the innocuous doorbell. Generally speaking it's just there and it just works - you push the button and the doorbell rings (see below for actual technical explanations of how doorbells work). But what to do when the doorbell stops working? You've got two options: repair or replace...
Sites such as This Old House, HowStuffWorks, eHow suggest first checking the button, which is the usual culprit as it sits outside exposed to all sorts of nasty weather (and abuse from the neighborhood kids always playing Ding Dong Ditch). Open up the casing and unscrew the two attached wires. Cross the exposed wires and if the doorbell rings then you know the button is at fault. Simply find a new button from your local hardware store and screw things back together and you should be back in business.
If the chime doesn't ring, the aforementioned sites will walk you through further testing of the chime assembly and transformer, which we will defer technical expertise to them on. Generally it'll be a problem with the button though, and if not then it may be easier to just go with a fancy new doorbell. Hey, we'll take any excuse we can to buy new toys.
Unplggd's Down Right Awesome Doorbells presents several options for upgrading your doorbell in fun and funky ways.
Doorbells: An Often Overlooked Part of Home Decor brings an updated look to your old doorbell.
Go high tech with D-Link Doorbell Allows Front Door Monitoring via 3G.
Get an early warning before they reach your door: Driveway Patrol Keeps Watch So You Don't Have To
A search on Google for "wireless doorbell" brings up thousands of results, and they can be really inexpensive at under $10 for the cheapest kits, although quality may (or may not) be an issue.
Great resources for explaining how a doorbell actually works:
(Images: Flickr member //www.flickr.com/photos/atomicjeep/13811301/" target="_blank">atomicjeep licensed for use under Creative Commons)