The above guy is partially right, except that lagers can have residual sugars, just try a dopplebock or a good malty marzen.
Lagers are brewed with a yeast that sinks to the bottom of the fermenting tank, and is more resistant to low temperature fermentations. Ales are brewed with floating yeast that ferments at the top of the tank, and is more comfortable above 60degrees F. However, lagers can be fermented warm and ales fermented cold depending on what specialty style of beer is being made.
Beer is now a generic term that usually refers to most grain based (non-distilled) alcoholic beverages. Technically, even Sake is a beer, since it is made from grain.
However, Ales were traditionally unhopped... beer or more accurately bier was used to describe hopped fermented beverages. The bittering agents in ales were typically herbs, flowers, grasses, and woods.