There are over 2 dozen different styles of beer in the world. However, all these styles fall within two types, Ales and Lagers. These two types are different because of the two different kinds of yeast used in the fermentation process. 

Ales

Ales are "top fermenting" beers. They are made with a top fermenting yeast called Saccharomyces cerevisiae. This yeast clings together in large clumps near the surface during fermentation. Ale yeast works best at higher temperatures and, as a consequence, works more quickly to convert sugars into alcohol, but does not convert all the sugars and leaves a sweetness in the beer. Ales are more cloudy with flavors that tend to be louder than Lagers and that change, even improve, with warmer temperatures. As your Ale warms up in your glass it gets more bolder tasting. Darker ales are even more flavorful.

Lagers

Lagers are "bottom fermenting" beers. They are made with a bottom fermenting yeast called Saccharomyces uvarum. This kind of yeast does not clump together and therefore sinks to the bottom during fermentation. This Lager yeast works best at low temperatures and slowly ferments more sugars into alcohol. So while Ale yeast actually converts sugar faster, Lager yeast converts more sugar over the longer fermentation. Therefore, Lagers tend to be clean, crisp and less sweet than Ales. Darker lagers are more malty tasting than light lagers, but both share a common crispness. Lagers are usually served very cold and the flavor typically will not improve with increasing temperature.

Most lagers consumed today are of the variety called Pilsners, which was first brewed in the town of Pilsen in 1842. The original Pilsner Urquell is still one of the best beers in the world.

The different styles of beer are differentiated by several elements that go into the brewing of each beer.

They include: appearance (which in itself includes several factors like color, head, clarity, etc); aroma; flavor; strength or alcoholic content; ingredients, such as the type of grain used, the variety of hops (and they way they are toasted or roasted prior to the brewing process itself) and, of course, the type of yeast used.

Some of the most popular styles are: Pale Ale, Blonde Ale, Amber Ale, Brown Ale, Cream Ale, Old Ale, Stout, Bitter, Pilsner, Steam Beer, Wheat Beer, Pale Lager, Ice Beer, Malt Liquor and American Lager.

Drink your lagers cold. Your ales should be drunk only slightly cool. Of course, that's only my opinion.