"Real ale" is a term used in
England for the distinctive kinds of
beer that are really only found in England, very popular within the country, and not very popular outside it.
Beer is the generic term for everything from Budweiser Budvar to Guinness Foreign Extra, but in an English context the difference between "beer" and "ale" is fiddly and a historical curiosity. Anyone who tells you they're fundamentally distinct is (a) bluffing, (b) a pedantic bore, or (c) both.
What is important is the difference between proper English beer ("real ale" or "bitter") and the rest of the world's. Real ale is alive. It continues to ferment in the barrel even as it's being sold. So it changes from day to day (an 18 gallon barrel should last three days or so), and the first few pints from the beginning or the end of a barrel will taste differently. Typically, the beginning is watery but palatable, the end can be quite nasty. If real ale isn't sold fast enough, the fermentation goes too far and it begins to develop aldehydes, making it taste of rotten apples. This is not desirable.
You cannot walk into a pub and order "a pint of..." (Burton, Bass, Young's, Ruddles, whatever) and get the same thing. It is not like lager, which is more or less the same everywhere. It depends hugely on how much care the pub landlord takes of the beer, what day it came in on, what volume they sell, what temperature the cellar is at, and so on. You really need to test and get to know the individual pub.
The other key thing is that real ale is on a hand pump. It is not under pressure: whereas lager and keg beer have a filling of carbon dioxide, real ale has to be pumped up the lines by hand. The quality of the head is affected by this. Two big pumps may be too few to give a good head on a pint.
English beer is not served at room temperature. It needs to be slightly cooled, at around 13°C (some CAMRA expert may correct me here). Before you start on jokes about English climate, bear in mind that the pub is warm, very warm: ideally it should have a roaring fire in winter. (A real fire.)
Real ale is nutty, subtle, mellow, bitter, sweet, and delicious. People often convert from lager (in the ignorant days of their youth) to real ale. No-one ever converts in the opposite direction. A good real ale is one of the great blessings of England.
Strength
A word of caution for the beginner. There is something peculiar about the
alcoholic effect of real ale. It gets
much stronger as the alcohol goes up. Almost all such beers are between 3.5% and 5.0%
alcohol. Now 5.0% might not sound much, but it is. Beware! You will be on your ear after a few of these.
Consider a pint of beer. That's 570 mL, compared to a 750 mL bottle of wine, which might be 13% alcohol. Now you'd expect two pints of 3% beer, one pint of 6% beer, and a large glass (a little under a third of a bottle) of 12% wine, all to have the same potency, because they have the same amount of alcohol.
Well you can have your glass of wine and barely notice the alcohol. It's pleasant, but doesn't affect you. Beer at only 3% is too weak to be worth drinking under any circumstances: it would have no effect, and the second pint is just wasting your money on still no effect. But beer at 6% is very dangerously strong. You would notice a powerful effect straight away, and you would be, if not actually drunk after the end of the pint, certainly in no fit state to drive. DO NOT HAVE a second pint of 6% unless you already know the beer. In fact even 6% is so strong that (a) there's possibly only one beer in the country that strong that's served in normal pint glasses (Owd Rodger), (b) you wouldn't want to anyway, and (c) some pubs mightn't even serve it to you unless you were a local. The strongest that is normally* served in pubs is 5.0%, or 5.5% tops. Do not toy with this. Several pints of this will wreck you.
I don't know why there is this disproportionate effect compared to the nominal alcohol content. It's not just the things that give you hangovers the next morning, it's the immediate effect as you taste it and it goes to your head. From 3.5% to perhaps 4.2% is weak and you can indulge all evening (it's a "session bitter"), around perhaps 4.3% to 4.6% it's middle-range and you need to stop drinking them at some definite point, and above that you need to be aware in advance of how many you can handle.
Ales do get much stronger in bottles. I don't know what the chemical difference is, but you can get a number that are around 7% or 7.5%. Somehow these don't seem proportionately as bad as the very strong draught beers. Incidentally, the record for any kind of beer in bottles seems to stand at an astonishing 25% for the American variety Samuel Adams Utopias.
Other beers of the British Isles
I should mention that
Guinness is not an English beer, is not a bitter, and is not classed as a
real ale, though it's a
kissing cousin. I notice American tourists often enthusiastically order Guinness over here, and I think they should be trying the real thing instead: wait till you're in Ireland before having stout.
Also, a warning. Would Scottish readers mind looking away now, please. Okay, Scotland makes things basically similar to England's real ale, except that they're horrible, shockingly sweet things. In Scotland, drink the whisky, or find an English beer.
Finally, Wales isn't much renowned as beer country, though it does do some perfectly good lines.
* Lady_Day has seen one called Old Tom served in a pub, at an astonishing 8.5%.