Ginger beer is the term for two very different carbonated beverages, both flavored heavily with ginger. Ginger beer was first made in England in the mid-18th century, and was produced by fermenting ginger, sugar and some sort of acid (historically cream of tartar -- tartaric acid, but usually citrus juice today) with a unique symbiotic mass called a ginger beer plant. This type of ginger beer is alcoholic, and can reach 10-12% alcohol, though most are under 5%.
The ginger beer plant itself is made up of Saccharomyces pyriformis, S. cerevisiae (common brewer's yeast), Mycoderma cerevisiae, and the bacterias Lactobacillus hilgardii, Acetobacter aceti, and a few other strains that can vary based on where the coolny is grown. Most of these are weak fermenters, and each ferments and goes dormant in turn, leading to an interesting fermentation curve. Acetobacter produces acetic acid (vinegar), which means that ginger beer brewed using this method tasted similar to Belgian lambic beers.
Commercially produced ginger beer is much more simple. It is effective a forced-carbonated soft drink flavored with ginger, given acidity with citrus (which can vary from brand to brand), and sometimes made spicy though the use of capsaicin, the active ingredient in chiles.
The main difference between ginger ale and ginger beer is that ginger ale is lighter, drier, and less ginger-forward. Ginger beer tends to be heavier, richer, and sweeter, with a noticeable ginger flavor that is sometimes enhanced with capsaicin. Ginger beers are often cloudy, or have small pieces of ginger remaining in the bottle.
Some common brands of ginger beer include Barritt's, Reed's, Ginger People, and Cock & Bull, though there are many regional variants.