Spoilers for X-Men: Days of Future Past follow.
At the end of X-Men: Days of Future Past Wolverine wakes up in the year 2023 to find that the war with the Sentinels never happened, the dark future we see at the beginning of the film has been erased and - what's more - the events of at least X-Men: The Last Stand never occured, as Jean Grey and Scott Summers are alive, well and teaching at Xavier's School for Gifted Youth. Also alive and present: Rogue, Iceman, Storm and Beast, in his blue, furry form. Kitty Pryde and Colussus are there, and they might even be a couple. That's about all we know - we don't even see Wolverine pop his claws, so it's unclear whether he has his adamantium.
There have been a couple of articles that have attempted to find continuity holes in the new film - like this one from Slashfilm, which doesn't quite grasp the difference between a continuity error and a skipped beat (ie, Professor X being back in his body isn't a continuity error, it's just unexplained) and this one from io9, which seems to misunderstand basic character stuff (like Magneto, full stop). The problem that neither article takes into account is that there are no longer continuity issues in the X-Men films because the previous X-Men films (except First Class) never happened.
Yes, not one of the previous X movies - not X-Men, not X2, not The Last Stand and not The Wolverine or X-Men Origins: Wolverine - happened anymore. You can throw those movies away. This is very familiar stuff to comic book readers - we've been experiencing these sorts of reboots since Crisis on Inifite Earths in 1985 - and fans of Star Trek, who saw a similar time travel gag wipe away the entire continuity built up over 50 years. The argument for Trek can be applied to X-Men - because one time traveler remembers the old continuity it technically happened - but they've been using that out for a long time in comics too. The Psycho Pirate remembers the pre-Crisis continuity (or maybe he doesn't since DC recently undid all of that continuity in the New 52). The effect is still the same - those stories never happened.
Both of the linked articles make two profound mistakes: one, they don't understand that at the end of Days of Future Past everything we don't see in the school is gone, and two they don't understand the idea of a retcon.
Retcons and reboots are prevalent in comics but they're different. A reboot is what we saw in this movie, a do-over, a wiping of the board. A retcon is a stranger animal, and it's when a writer attempts to either fix a continuity error or unwrite a disliked story element by revealing missing history. One of the things complained about in the io9 article is that Mystique and Professor X should have a different relationship in the original X films because of their tight personal history, as revealed in First Class. And the article is right - the way Professor X and Mystique interact makes no sense. But that doesn't make it a continuity error yet - a writer could have inserted a plotline where one or both of their memories are wiped. The timeline between First Class and X-Men is long enough that any number of stories could happen which might explain away this seeming inconsistency. If there had been no reboot and the movies had continued moving forward to finally meet up with X-Men chronologically and that relationship hadn't been fixed then, yes, that would have been a continuity error.
That gets tougher when you have issues like Bolivar Trask being both Peter Dinklage and Bill Duke, actors who are very, very different physically. But the reboot that happens halfway through Days of Future Past - when they stop Mystique from killing Trask and reveal mutants to the public decades early - takes that issue off the table. And by the way, there's no reason why it couldn't be revealed that Duke's Trask was just the son of Dinklage's, just as the two seemingly incompatible Moira MacTaggarts could have just been relatives (or bizarrely similarly named unrelated people). But it doesn't matter anymore, because only one Moira currently exists in continuity, and that's the one from First Class.
At this point the only continuity issues that can exist will be those between First Class and Days of Future Past. There's a shot of a kid who could be Cyclops in Cerebro in FIrst Class, which would make end-of-the-movie Days of Future Past Cyclops about sixty years old, but that's so easy to hand-wave away it isn't even worth addressing. That stuff is more problematic if X-Men Origins: Wolverine is still in play, but it isn't, so who cares. (There's one other thing to take into account: X-Men takes place 'in the near future,' so timing a lot of stuff to 2000 doesn't work. The 2023 continuity - and this is canon, as the movie establishes the future is 50 years after 1973 - could be taking place just a handful of years after The Last Stand, as opposed to the sixteen or so years in reality. Yeah, it's a little confusing)
In a lot of ways all of this makes the X-Men films the single most comic book-y comic book movie series; continuity in a shared universe has always been a pain in the ass because writers want to tell the stories they want to tell while fans want the stories to connect into a cohesive whole. DC was particularly susceptible to this because it had so many decades of stories and characters who were being contradicted left and right by writers who thought they were just writing entertainment for kids, not semi-religious tracts.