Note: Phil touched on this in his excellent review of Skyfall, but it's my site and I wanted to talk about it too!
Every since George Lazenby, the first replacement Bond, said "This never happened to the other fellow" in On Her Majesty's Secret Service, there has been a theory among Bond fans that perhaps James Bond isn't a guy, but rather a code name assigned all the 007 agents. It would certainly explain why he's so free in giving out his moniker, and it opened the door for a fantasy all-star Bond team-up, a movie where previous Bonds came out of retirement to help the current guy.
Of course anyone paying attention to the films themselves know this is simply not true. The best evidence, until Skyfall, has been Tracy Bond, who actually married 007 in OHMSS. But it was a Bond film and, even though she was played by the wonderful Diana Rigg, Tracy had to die. Blofeld, who would be a Bond villain throughout a number of movies (played, over the course of the series by the likes of Donald Pleasance and Telly Savalas), had her killed in a drive-by shooting.
Tracy's impact on Bond did not end there, though. Instead of wiping her from the memory bank like so many other doomed Bond girls, our secret agent mourned her over multiple movies. She's mentioned in Roger Moore's The Spy Who Loved Me, four movies, two Bonds (Connery briefly came back in between) and eight years later. That was a circumspect mention - someone asks Bond if he was ever married and he gets all PTSD about it - but For Your Eyes Only makes it really explicit. It opens with Bond at Tracy's grave and then getting his revenge on Blofeld (who, hilariously, is not named for legal reasons). He dumps Blofeld into an industrial smoke stack. The end!
So we've proven that George Lazenby and Roger Moore are the same Bond, as they were married to the same dead woman. But what about everybody else? It's less explicit in later films, but Tracy's ghost is hanging around. In License to Kill it's mentioned that Timothy Dalton's Bond was married once, long ago. Pierce Brosnan's Bond doesn't have such an obvious Tracy moment, though Bond fans do believe she's mentioned in passing in two of his films. In Goldeneye villainous 006 asks Bond if he's found comfort in the arms of many women to help him forget the ones he failed to protect. That's really vague, especially since so many Bond girls have died. You can read this to assume that the deaths of these random women meant more to Bond than he let on, or you can assume 006 is talking about Tracy. Still vague, but less so, is The World Is Not Enough, where Bond is asked if he ever lost anyone he 'truly loved,' and he gets uncomfortable and avoids the question. None of these are on the level of For Your Eyes Only, but the hints are there.
Still, the hints are vague enough that they can be ignored, and we can all pretend that Lazenby is just Moore, and that everybody else is a different guy in the Bond identity. Until Skyfall.
Mild spoilers for Skyfall follow.
At the end of Skyfall, Bond takes the battle with supervillain Silva to his own secret lair: the family mansion in the Scottish highlands (called Skyfall). There we briefly see the family cemetary, where we see the headstones of his parents, Andrew Bond and Monique Delacroix Bond. And so it is established, fully and once and for all, that James Bond is actually a guy named James Bond. It is not a name that is conferred by MI6, it is not a secret identity. It is simply a shittily-hidden actual identity. All of the previous James Bonds were definitely the same guy, and all future post-Craig Bonds will still be the same guy.
It's worth noting that Bond's parents are canon from the books. In You Only Live Twice, there's an obituary for Bond written by M (echoed in Skyfall, by the way) and his parents are mentioned as being deceased. They were killed in a climbing accident, which I do not believe is mentioned in Skyfall. HIs aunt is mentioned in the film, though.
And so you have it. With a quick shot of a headstone one of the biggest continuity questions about the Bond series is cleared up. There will never be a movie where Roger Moore or Timothy Dalton swoop in to tell the current 007 how things were done in their days. It's just one guy, and it's always been just one guy.