Beer is essentially
fermented
barley. I suppose you could do it with other
grains and produce pretty similar drinks -- the lovely
Hoegaarden, for example, is called
wheat beer, though I don't know of any drink that uses
wheat instead of barley. The German
Weizen beers are no more than 60% wheat.
Anyway, minor variations aside, beer is the plant barley which has been allowed to ferment, i.e. turn into alcohol by a process of microbial decay. If you distil barley, a different process, you get whisky. But no-one calls whisky beer. The words beer and barley and brew may be etymologically related; - or it might be an early Germanic borrowing from Late Latin biber 'drink'.
The ancient Egyptians made beer; their name for it was h.nqt (which you can pronounce heneket). The ancient Greeks called it zythos. (The modern Greek is the borrowed word birra.) In old Germanic it was called ale. Up to about 1400, England drank ale, not beer.
Two inventions arose on the Continent about that time. One was laying it down (lager) in a cellar and keeping it cool. The other was the addition of the plant hops, which gave it a bitter flavour. This innovation was much resisted by the English, but it was so much cheaper that Henry V supplied his troops at the siege of Rouen with it, brewed in London, at half the price of good old English ale, in 1418.
This new drink, ale made bitter by the addition of hops, was called beer. That is the actual historical distinction in English, though it is not how the two words are now used.
Old English ale, a strong, sweet barley drink without hops, survives to this day in the minority-taste bottled confection called barley wine. It is immensely strong and will poleaxe you if you drink a lot not being used to it. But that is what (supposedly) the monks of the Middle Ages lived on when water was full of deadly germs.
Beer is now the generic term. It covers lager, the German (and American and Australian) freezing-cold high-pressure drink; it covers stout, the thick black, vitamin-rich drink of Ireland; it covers "bitter", the commonest variety of English beer (see real ale for how this differs from others); and it covers intermediate varieties like "mild" (sweeter and weaker than bitter), "porter" (strong and thick, intermediate between bitter and stout), and Belgian lambic, and "wheat beer", where something other than barley is used in the process.
The brewing process itself is quite complicated and I don't know enough to discuss it intelligently. It starts with "malt", which is barley grains fermented in the air. See mneek on ale for a good explanation.
In England, real ale snobbery discussions sometimes use the word "beer" to mean "English beer" i.e. "bitter" i.e. "real ale" i.e "not that lager muck that you're drinking" -- but I wouldn't take this seriously. Beer is the most generic term, and even in England it covers all varieties.
Oh. For the record. I like beer. A lot. Real beer, that is. :-)
Thanks to Gone Jackal for correcting my memory of the Egyptian, and introducing doubts about the Greek. In fact neither the printed Liddell-Scott nor the Perseus on-line Liddell-Scott-Jones is quite satisfactory here, so I don't know what the truth is.
Thanks also to Gone Jackal, bn557, PaulM, and Blue_Bellied_Lizard for discussion on wheat beers.
and to hollowboy for pointing out my slip in that the fermentation process is fungal (with yeast), not bacterial.