In Micronauts #1 (by Bill Mantlo and Michael Golden), Marvel did a comic book based on the Mego toy line of the same name, which was, in turn, based on the Japanese toy line, Microman. The toy series was about rebels fighting against the evil Baron Karza in the Microverse. Mantlo introduced a bunch of new Microverse heroes, including the lead of the book, Commander Rann , who woke up after 1,000 years to help lead a rebellion in the Microverse against the evil Baron Karza...
Since Rann (and the other heroes, like Marionette and Bug) were Marvel inventions, Marvel is able to continue to use them even after they lost the license to make Micronauts comic books (the toy line only lasted a year, but Marvel kept making the comic book until 1986!). As Marvel doesn't own the rights to the Microverse itself, though, they couldn't use it in their movies, so they came up with the name "Quantum Realm," instead. I believe the science adviser for Ant-Man coined the name.
The Micronauts were a pretty popular series for a while there, even leading to an X-Men/Micronauts crossover miniseries where Charles Xavier's evil side tried to take over the Microverse (Charles Xavier has a dark, dark side to himself). For a moment there, Xavier's dark self even took over Xavier's body and then totally perved on Dani Moonstar. Yikes, this is unsettling.
Once the Micronauts ended, the Microverse would pop up just on rare occasions, either when people were fighting the Psycho-Man or when they were teaming up with the Micronauts characters that Marvel still owned (they couldn't call them the Micronauts, but the basic point got across).
In 1999, Peter David cleverly began using the Microverse as the new area where Rick Jones and Captain Marvel would switch bodies when they switched bodies with the Nega Bands (in the original comics, the one who was not on Earth was stuck in the Negative Zone).
Back in 2013, at the end of his run on the Avengers, Brian Michael Bendis had the Avengers travel to the Microverse to rescue Janet Van Dyne (which is eerily similar to the upcoming plot of the Ant-Man and The Wasp movie, right?) who was trapped there ever since she seemingly died in Secret Invasion...
Beyond that, there are always occasional stories set in the Microverse, like a crossover between Venom and Scarlet Spider that saw Carnage escape into the Microverse. It's a regular part of the Marvel Universe and it seems like it is set to be a major part of the Marvel Cinematic Universe, as well!