Uhh, budweiser is a lager. There are really only two types of beers by brewing types. Ales and Lagers. One is brewed warmer than the other. It's two different temp ranges at which the beer is brewed at. A "pilsner" is a lager brewed beer that uses a light malt and cold storage standards.
Basically since the beer purity laws of Europe, beer is basically made with 4 main ingrediants. Malt, hops, yeast, and water. It is either brewed as an ale or lager. That's it. In older times, there were other things that were used instead of hops to cut the sweetness of brewed beers.
Despite that, there are variance in malts used, hops used, fermentation length, storage differences, and filtration. All that makes a difference to the final product that determines the "beer" category it fits in.
Ales vs. Lagers are more than just brewing temperature, one yeast is bottom fermenting (lager) vs. top fermenting (ale). Lagers also have to be aged (or, lagered) for some time so off flavors can dissipate.
Also, that's a German law, not a European law. A lot of other brewing cultures utilize fruits and spices in their beers, only in Germany is it actually illegal.