In Britain, bitter (bitter ale in modern usage, pace santosvega) is so called to distinguish it from mild (mild ale), a less hoppy (and generally darker), but otherwise similar form of beer. In modern British usage, they are both called ales to distinguish them from lagers, the types of beer traditionally preferred in most European countries, and the USA.
Even many British people are probably not aware of this, however. Growing up as a teenager in the late 1960s and '70s in the south of England (and spending much time in pubs), I was not aware that mild existed, and simply understood “bitter” to be a word for British style draught beer, as opposed to continental style lager. Bitter was the default. If you asked for “beer” you would get a bitter. If you wanted lager you would specifically ask for that. (We knew the word “ale,” but anyone who asked for an ale would have been thought to be either a pretentious git, or else a Northerner, probably even a Geordie.) No-one ever asked for mild, probably because most of us did not know it existed, and I am reasonably confident that very few, if any, of the pubs round my way even stocked it.
I did not discover the existence of mild until I went to university in a city in the north of England, and, even then, only after I ventured out of the student areas and the city center and visited some pubs in the working class parts of the town. There, you could still buy mild on tap, although I think bitter and lager were still more popular, and mild was mainly a drink for the older generation. Some people, I discovered, liked to drink mild and bitter, a mixture of the two (also known as half and half, although that could have other meanings, even within the context of beers). Apparently, mild used to be widely drunk all over Britain until the 1960s, when, for some reason its popularity crashed, and it all but disappeared except, as I found, in working class areas in the north.
I cannot speak to more recent times, as I have not lived in Britain for twenty years, but according to Wiki, mild has recently made a little bit of a comeback. On the other hand, I have also heard that more lager than bitter is now drunk in Britain, although bitter (unlike mild) is still certainly universally available and widely drunk.
Some bitters, really are pretty darn bitter. One I remember is Greene King Abbot Ale, which some of my friends loved, but I always regarded as vile. Boddington’s (as I recall it) is pretty much at the other end of the scale of bitters. I think it might almost count as a mild if only it were a bit darker and heavier.