How many editions of Dungeons & Dragons are there?
As a generic answer for the non-hardcore fan, I think most people would define the following:
Certainly you could split it further (the various Mentzer/Moldvay differences, etc.) , but those are the major divides most gamers would likely note.
Of course, no edition of D&D survives contact with a gaming group intact; if there are 40,000 D&D sessions this weekend, there will be just about as many “versions” being played.
Dungeons and Dragons is a game designed by Dave Arneson and Gary Gygax; there was only one edition. Gygax went on to write Advanced Dungeons and Dragons, of which there was also a single edition.
Various companies have printed game rules with the same name.
Edit: there's been some unhappiness with this answer so I'll expand it a bit. "Dungeons and Dragons" is a trade name which has been owned by various companies. As a trade name it can legally be applied to a box of cereal or a car or, more likely, a game of fantasy miniature combat or a simple "Dungeon!" style board game. In other words, it does not designate any specific game any longer.
What it originally designated was a role-playing game by Arneson and Gygax. While later applications may change, the originality of that first application is set in stone - that game, consisting of the "little brown books" will always be the original sense of the name. A similar argument can be made for the use of "Advanced Dungeons and Dragons".
So, the question is whether other uses are the same game or something else with the same name? Legally, they are different games - this was the core argument for separating Gygax from TSR-era game design royalties. Morally, neither Arneson nor Gygax voluntarily gave up their control, they were forced out at different times.
So, neither legally nor morally can other games with the same name be said to have any more connection with the original game than any retro-clone or other product with the name "Dungeons and Dragons" that I or anyone else might design. All they have is legal protection from the name being applied to other items, no matter how close to the original rules those other things may be. In that sense, which is actually quite a wide sense, those games are not, and cannot be, new editions, they are simply new product lines by the owner-of-the-moment who may apply it tomorrow to a themed tiddly-winks set.
The vast gulf between the original game and the so-called Fourth Edition demonstrates this pretty clearly - the name is meaningless as a way of determining whether something "really" is Dungeons and Dragons; it's just a marketing term.
Thus, there is only one edition of Dungeons and Dragons - the original - and one of Advanced Dungeons and Dragons - the original. Indeed, it's not clear what "fifth" edition is supposed to be the fifth edition of, since it's not actually called "Advanced Dungeons and Dragons". If the original creators had issued new rules, then they would have been genuine editions at least in the moral sense.
Being a bit nitpicky... and using a "The rules are different enough that they characters have different capabilities and/or limits on advancement" meaning for edition, I'd say:
Not D&D but precursors to it:
Putting these into families:
And that's without retroclones, pseudoclones, and knockoffs.
Note that, to me, a game edition can be any of the following:
Thus, the gaps between:
The Acaeum details the various editions and printings. It's well worth an extended browse just for the nostalgia of the cover scans. I'm going to say nine editions: OD&D, Holmes, Moldvay, Metzner, AD&D, 2e, 3e, 3.5e, and 4e.