There are two primary types of beer, Ales and Lagers. The primary distinction is the temperature at which the beer is fermented. Ales are fermented at higher temperatures 65-75°F, and Lagers are fermented much colder at about 46-55°F.
The second distinction is the type of yeast that is used in the fermentation process. Ales generally use top fermenting yeast. This means that the yeast floats on the surface for the first few days and then settles on the bottom. Lagers use bottom fermenting yeast, which does not float to the surface before settling.
There is a third type of beer that far less common than Ales or Lagers, and that is the Lambic. True Lambic is only brewed in the Payottenland region of Belgium. In Ales and Lagers the yeast is specially cultivated for the fermentation. Lambic is fermented by wild yeast. This means that the beer is exposed to the Belgian air which contains wild yeast and bacteria.
Among Ales and Lagers there are many, many different beers. See the beer articles for in depth information on many of these beers. For now the hierarchy of many types of beer is shown below.