Though it's a world where mutants exist and are hunted and feared by the human population that seems familiar to X-Men fans, the exact political landscape of The Gifted doesn't quite fit with anything we've seen before in the X-Men movies. The new series on FOX was created by Matt Nix (Burn Notice) and centers around the Strucker family, who are on the run and find refuge with the Mutant Underground, but it doesn't appear to exist in a shared X-Men universe in the same way as Marvel TV shows like The Defenders on Netflix do. The easiest way to look at it is The Gifted exists in an alternate timeline (like Logan), though there are various nods to elements from the X-Men movies.
In The Gifted's universe, the X-Men and their evil counterparts the Brotherhood are "gone". What happened to them hasn't been revealed, but the world as a whole seems well aware of the mutant superhero team. The X-Men certainly existed and their exploits: "mutants fighting other mutants" - as Reed Strucker (Stephen Moyer) described past events to his wife Caitlin (Amy Acker) - led to the creation of harsh new laws against mutants, as well as the advent of Sentinel Services. A federal agency with the seemingly unchecked power, under "the revised Patriot Act," to hunt down and detain mutants by any means necessary, Sentinel Services are basically a new incarnation of the black ops military run by William Stryker (Brian Cox) in the X-Men movies. Instead of a facility in Alkali Lake, which may exist in this universe, if Wolverine (Hugh Jackman) does, Sentinel Services has Mutant Detention Centers to incarcerate mutants.
The Sentinels themselves exist in a different form. We don't know if Bolivar Trask (Peter Dinklage) ever created giant mutant-killing robots in 1973, but a version of Sentinels in The Gifted appear as spider-like robots that are difficult (though not impossible) to destroy. Sentinel Services also utilize drones to patrol the skies and track down mutants. While not as comic book-y as giant robots, these new versions of the Sentinels are more insidious and no less fearsome to come across for mutants and humans alike.
The Gifted features a host of new mutants that haven't appeared in the X-Men movies, though Blink was featured in the post-apocalyptic future scenes of X-Men: Days of Future Past. Considering the Blink of Days Of Future Past (Fan Bingbing) was roughly the age of The Gifted's Blink (Jamie Chung) in the future, this is most likely a new version of the character made possible after the X-Men altered the timeline and/or created all new timelines in Days Of Future Past.
The biggest link to the X-Men movies in The Gifted is Lorna Dane, a.k.a. Polaris (Emma Dumont), a mutant master of magnetism who is acknowledged as the daughter of Magneto. Magneto has fathered at least one other mutant in the movies that we know of, therefore Polaris has (or had) a half-brother in Quicksilver (Evan Peters). Polaris doesn't as yet seem quite as powerful as her father has shown himself to be, but she is dangerous enough to be held in a plastic prison by Sentinel Services, just like the ones the movies have used to keep Magneto in check in X2: X-Men United and Days of Future Past.
All in all, The Gifted is staking out its own version of the X-Men universe that echoes the themes, incorporates familiar iconography and even showcases similar characters from the movies while forging its own path. By more strongly evoking bonds with the movies, however, The Gifted goes a step further in flaunting its X-Men ties than its sister series Legion on FX. Legion is a more original, twisted and psychologically daring tale of mutants very much set in its own universe, though its lead character David Haller (Dan Stevens) is the son of Charles Xavier.
Part of the fun for X-Men fans, however, will be trying to spot all the movie references The Gifted will be delivering and determining just how much the series does relate to the movies. As for whether any of the actual X-Men will appear in The Gifted, Matt Nix has stated the series will explain what happened to the famous mutant super team, so big name X-Men characters popping up in series remains an exciting possibility.
The Gifted airs Mondays at 9pm on FOX.