A lot of works use these terms interchangeably, though they have distinct meanings in reality.
A "plane" is technically a flat, two-dimensional surface, but in fiction this is used metaphorically: imagine reality as a whole to be a 3-D "space" with a bunch of 2-D "planes" layered on top of each other. Objects on one plane can interact with objects on the same plane, but not with objects on another, but it is possible for some entities to move between them or even extend across them. In fantasy this is used as an analogy to imagine 3-D "planes" layered "on top of each other" in 4-D "space". It can mean that each location in one plane corresponds to a specific location in another plane, but this is not always the case.
"Dimension" is basically always used wrong in this sense. A dimension is technically a pair of opposite directions: Length is the first dimension, width is the second, height is the third. (Ignore "time" for now; while "time" is dimension-like in some ways it works differently in practice.) Now imagine a fourth spatial dimension perpendicular to the other three. By traveling along that dimension, one might wind up in another 3-D space (same concept as the "planes" metaphor). So you're not really traveling to "another dimension", you are traveling through another dimension (direction) to reach an inaccessible place. But fiction writers often use "another dimension" to mean "another universe" anyway.
"Realm" literally means "a kingdom". It is much more common to use this in fantasy works where each region of existence is ruled over by its own god or cosmic entity or something; you'll rarely see this used in more scientifically-flavored works but it works well to give your work a more mystical flavor.
"Reality" suggests an additional layer of division between the worlds, it's not just an inaccessible place you can only arrive at by traveling along a fourth dimension, but rather from the perspective of one world the other world isn't even real. You'll find this when crossing over to mutually-exclusive continuities, or where the basic "rules" governing that reality are different from one to the next.
But...in reality, people just use these terms however they feel like. I'd advise against using "realm" in sci-fi works, and not to use "dimension" to refer to a universe at all, despite the fact that many writers do. If you want to be clever and modern, you can use "brane" - this is the term string theorists use to describe cosmic "spaces" regardless of dimensions involved. Brane is short for "membrane", evoking the same basic imagery as "plane" above.