Champagne is a "sparkling wine" that comes (of course) from the Champagne area of France. Three grapes can be used to make Champagne: Chardonnay, Pinot Meunier and Pinot Noir. It is produced by a technique known as Methode Champenoise.
In Methode Champenoise, there is more than one major fermentation. The first fermentation takes two to three weeks. The wine is then placed in very sturdy bottles (to withstand the internal pressure that will be part of the process) along with sugar and yeast (Liqueur de Tirage). A temporary cap (just like the type you find on a bottle of beer or soda) is placed on the bottle. The sugar and yeast cause a new fermentation to occur. Since fermentation produces carbon dioxide (the same gas that makes the bubbles in soda), which can't escape from bottle, what you get is carbonated wine. This fermentation also creates new sediment, which must be removed. This is done by placing the wines on their sides on racks at about a 45 degree down facing angle. Then every day the bottles are turned a bit (called "riddling" or "remuage"), and eventually also tilted farther down. After about 6 or 8 weeks, the sediment has now moved to the neck of the bottle, which the vintner then freezes. The bottle is opened and the force of the pressurized wine pushes the frozen sediment out of the bottle (this is called "disgorgement". Since the bottle is now no longer full, wine and sugar (depending on what sweetness desired) is added. The bottle is then given its permanent cork.
Some say Champagne does not mature in the bottle, so you needn't bother cellaring it. Others argue that you may enjoy a little aging on some vintage Champagnes. Mostly, I think they're drunk quick.
The French discourage (to put it mildly) the use of the word "Champagne" for sparkling wines made (even in the same manner) elsewhere in the world. Also know that not all sparkling wines are made using the Methode Champenoise. For example, instead of carbonating the wine in the bottle and hand turning the bottles every day, you could put the wine into huge stainless steel tanks for the second fermentation. This will get you much cheaper carbonated--or sparkling--wine.
[For the Future: how to open a bottle of Champagne; styles of Champagne, Naturel, Brut, semi-dry (demi-sec), etc.]