As I understand it from the movies (although there might be a different explanation in the comics):
Planets and solar systems are the same as we define them in the real world. Same with the Galaxy as a whole, where most events happen even for the space based movies, but some specific locations may be in other galaxies.
The realms, or more specifically, the 9 realms as mentioned in Thor movies are only those planets which Asgard conquered or has some dominion over. They also share some connections that makes them converge on each other every few thousands of years as seen in Thor the dark world. They may be in the galaxy or elsewhere, their actual location is unknown or irrelevant. For example, Hela says in Thor Ragnarok that she never understood why Odin stopped at 9 realms and that she intends to rule the whole universe as queen of Asgard.
Dimensions are a little bit more tricky, they can be considered different universes of themselves or different realities that are still part of the same universe. The term isn't well defined in the movies. However, based on the ancient one from Doctor Strange she refers to other dimensions as part of the "multiverse", hinting that they are in fact different universes.
I think the correct definition of an "alternate universe" or "parrallel universe" (which the movies haven't shown us or mentioned but are referenced and visited in the comics) would be universes that are very similar to our own but with different contingent histories, but basically with the same laws of physics. Dimensions on the other hand are those universes that are so different that different laws of physics apply (such as the mirror dimension or the dark dimension that has no time in Doctor Strange)
