In the first Thor movie, we are first introduced to the Nine Realms. Thor mentions Nornheim as one of the realms, and in Thor: the Dark World he mentions Nornheim again. However, Nornheim is not a realm in the comics or mythology.
In the non-canon videogame Thor: God of Thunder, it is said to be where the Rock Trolls lived before they were banished to Vanaheim. While this is non-canon, there may be evidence to support this claim. Before Thor's coronation ceremony, there was a canon deleted scene where Thor and Loki talk. Thor mentions fighting through a hundred warriors during their trip to Nornheim. In the beginning of Thor: the Dark World, the Asgardian armies are fighting to restore peace in the Nine Realms, and while on Vanaheim they encounter the Marauder Korg, who appears to be a Rock Troll. However, this proves nothing, since the hundred warriors aren't specified and Nornheim is known for being inhabited by powerful magic users. As for Korg, the Marauders have spaceships to travel between realms, as well as the fact that Korg is actually a Kronan, ancient rock-like humanoids from the planet Ria. Rock Trolls are actually more human-like and are not made from rocks.
In the comics, Nornheim is a subsection of Asgard where the Norns live, as well as Karnilla, an Asgardian who has become an expert in sorcery. This has not been explored in the movies yet, although they could come up in Thor: Ragnarok.
In mythology, the Nine Realms do not include Nornheim. The tree Yggdrasil is watered from three wells: one on Asgard, one on Jotunheim, and one on Niflheim. The one on Asgard is the Well of Urd and is where the Norns live. This is very similar to the concept of Nornheim, a section of Asgard where the Norns live.
The idea of a well also ties in to Thor's side quest during the Age of Ultron, when he goes to the Water of Sights. Thor mentions that although there is a Water of Sights in every realm, they are all "reflections" of the true Waters of Sights. He does not say where the real one is, but it could be the Well of Urd on Asgard, or in the movies on the yet to be explored Nornheim.
This idea of Waters on every realm could be modeled after the other Wells. In mythology, Odin gave up an eye to drink from the Well of Mimir on Jotunheim, which is said to have given him his great knowledge and wisdom. This idea is similar to the Water of Sights, in which Thor gains knowledge about the Infinity Stones. While this could still be explored in the movies, the canon story arc is that Odin lost his eye fighting against Laufey.
Additionally, Thor was possessed by a Norn while in the Water of Sights, as shown in a deleted scene. This scene was cut due to time constraints and does not contradict the canon story line. This further supports the possibility that the Water of Sights is the Well of Urd on Nornheim.
We see little of the Norns themselves. One possessed Thor, but there are three Norns. They are not mentioned anytime else in the movies, and they are not even said to inhabit Nornheim. All we know about them is they allegedly know great knowledge and the future, and they feed on life, requiring a sacrifice to tell their knowledge.
While it is possible for Nornheim to be its own realm with nothing to do with the Norns, it makes more sense that it would tie in in this way. Or at the very least, it could be a different realm that has its own people, as well as the Water of Sights with the Norns hidden deep within it. Other things are possible: Nornheim being one thing while the Water of Sights and the Norns living on Asgard; Nornheim being one of the planetoids seen orbiting Asgard (which leads me to believe that the other is possibly Valhalla), or they might just do away with the Norns completely, since it was a deleted scene. Or they might just be there, like a thing that exists but we don't go into, like Dr. Sterns being The Leader in the Incredible Hulk movie. I'm interested to see where this will go in later movies.