How is beer classified into different types?
Can the types of beer be arranged in a hierarchy with "ales" and "lagers" at the top? Are there beers that do not fall under either category or some that fall under both?
What are the divisions under these categories?
What are the defining characteristics distinguishing one type of beer from another?
http://beeradvocate.com/beer/101/ has a vast amount of information on beer styles.
To answer your specific question, an ale is top-fermenting whereas a lager is bottom-fermenting. Lagers are generally fermented at lower temperatures than ales.
I believe that lambics would constitute a third category, since they are traditionally fermented by wild yeasts, but opinion may vary on this, I don't know.
There are dozens of styles of ales and lagers and a few different lambics, but the method of fermentation is the main distinguishing characteristic between the three types.
Another Beer Advocate link: http://beeradvocate.com/beer/style
Those folks make a three-level hierarchy:
You can see another structure emerging within theirs, however, in which descriptors are tacked on to a style. For example, you have an ale, a pale ale, an imperial pale ale, a double imperial pale ale, an american imperial double pale ale, etc.
The classification of different beer types comes from how they are made. The first distinction is top-fermenting vs. bottom-fermenting, i.e. does the yeast get thrown in on top and work its way down the mash, or is it the reverse.
In general, lager yeasts are more sensitive to temperature control during brewing and result in a subtler and cleaner flavor. Pilsners and most German beers are of this type. Ales tend to have a larger variety of yeast strains used, so don't have as consistent a profile as lagers. (There's exceptions of course, Schwarzbier is as dark and malty as they come.)
All other styles are a result of their ingredients, locations and histories. Most have a specific quality such as Labmics, which are a sour beer that uses spontaneous fermentation. Rauchbier (also known as Smoked Beer) have a distinctive smoky flavor and sometimes are actually smoked before bottling. (And, frustratingly, can be either a lager or ale before being smoked).
Beers that use Wheat as one of their malts tends to make another broad category of ales such as the Hefeweizen and Whitbier.
Not every category is so easy. You'll be hard pressed to find a beer expert who can quote the differences between a Porter and a Stout without speaking in generalities and flavor profiles. Similarly, since the naming of beers isn't regulated everywhere in the world what is one breweries "American Pale Ale" is an others IPA.
In order of lightness/colour, with the palest at the top
More here
There are more types, for instance, wheat and rye beers.See http://www.beeradvocate.com/articles/351/ and http://www.beeradvocate.com/beer/style/12/ .
