In general, beer can be classified into two main categories: ales and lagers. Both ales and lagers run the entire course of particular flavors. They both vary in a wide range of colors, bitterness, aroma, and maltiness. The main difference between lagers and ales, though there are more than a few, lies in the fermentation.
Two big differences are the type of yeast used and the fermenting temperature. Ales are fermented with top fermenting yeast which, as you can imagine from the name, hang around the top of the fermentor. Ales are generally fermented at warmer temperatures (about 65-75 degrees) for short periods and are very simple to make. Lagers, on the other hand, are fermented at cooler temperatures (about 50-55 degrees) for longer periods of time. Lagers use bottom fermenting yeast, which contrary to its ale relatives, hang out at the bottom of the fermentor. The result of this cooler, longer fermentation and bottom fermenting yeast is generally a crisper, clearer appearance many of us associate with lager beer.
This longer fermentation time plays into the very name "lager." The word comes from germanic roots, meaning "storehouse," referring to a place, often a cool place, where a person might keep food and drink for extended periods. It is thought that lagering may have actually been done by mistake as far back as the Dark Ages, when people would store their beers in cool plaes for later use. These beers happened to have airborne lager yeast float into them, and thus by dumb luck, lagers were invented.
Other differences exist as well. Beer-Faq has a good breakdown of this in chart form. Ale is an older style of beer than lager. Ales were first made in the days of yore, whenever those were, by brewers who knew that fermentation would happen if they left the tanks open at room temperature. It was only in the second half of the nineteenth century that Louis Pasteur figured out that yeast was responsible for fermentation. After that, people figured out how to culture yeast and make beers and wines with different characteristics. Since then there has been a new set of possibilities available for styles based on yeast differences, rather than other ingredients. People isolated and grew the yeasts that perform well at lower temperatures and used them in their best envornments.
Another difference is that ales can have stronger, more pronounced, and even aggressive flavors, while lagers are often a bit more subdued. In addition, lagers are normally served at a cooler temperature than their ale counterparts. The proper temperature range for a lager might be in the low 40's, while ales are usually served about ten degrees warmer, in the lower 50's.
Try doing a taste test between your favorite ales and lagers to see what other differences you note.