There are four or five competing theories for the origin of the word “beer” and, frankly, none of them is particularly convincing.
The same is true of the word “ale”, as it happens: despite “ale” and its sisters, such as öl in Swedish and alus in Lithuanian, being found in languages from Britain to the Black Sea via the Baltic, no linguist has any good idea how it originated, with some of the ideas put forward being way out there in the unlikeliness ionosphere.
Of the four “great” families of words meaning “alcoholic drink made from malted grain”, however, we can be reasonably certain about the origins of the other two, the Slavonic “pivo” group and what might be called the “cerevisia” group, after the Latin word for “beer”. (Or, to be accurate, one of the Latin words for beer, since as well as the spelling we’re familiar with in the name of brewing yeast, Saccharomyces cerevisiae, the word also occurs in various Latin documents in the forms cervisia, cervesia, cervese, cervesa and cervisa.)
Taking this “cerevisia” group first, the Romans, who were wine drinkers rather than beer brewers, nicked their word for beer, in all its spellings, from speakers of a Celtic language. The original Proto-Celtic for “beer” was probably something like *kormi (that asterisk is the etymologist’s symbol indicating a word that has not been attested, but whose form can be worked out on the basis of later variants), going back to an earlier Proto-Indo-European word *kerm- (that dash means there was an ending on *kerm but we don’t know what that ending was.) *Kerm looks to be the root of a few other words in the Indo-European family, such as Russian korm, meaning “fodder,” an old Slavonic word krma, meaning “nourishment” or “food”, and Latin cremor, meaning “broth”, or “pap”.