Before the 17th century ale referred to beer brewed without the addition of hops. Today the term 'ale' has a much less precise usage and may refer to beer made with unroasted malt, beer with milder hops flavor or beer made with quickly fermented hops, etc.
Ale is a fermented alcoholic beverage that is brewed from the same basic ingredients as lager beer. Different fermentation methods give it a different flavor. (Ale is made with a top-fermenting yeast and uses higher temperatures). Ale generally has a stronger hop flavor and higher alcoholic content than beer.
Horseradish was used to produce Horseradish ale in the 17th century.
Lager beer is the dominant beer style throughout the brewing world today, except in England. Ale is the primary style of beer consumed in England.