Separate regions of the universe, each of which appears to be a self-contained, expansive continuum of space-time with its own set of values for certain basic physical constants that define the form of matter that exists within them. These regions map on to each other and appear to exist within the same "space" but with differing "reality phases". The differences in reality phase create dimensional "barriers" that keep matter within different dimensions from interacting, making each dimension effectively invisible to other dimensions. Thus, objects cannot travel between them without the use of special technology, though information does seem to cross between them psionically.
Terminology and Classification
Though they had been speculated about as branes by Human physicists in the 21st century, dimensions other than TrueSpace were not known by Humans to exist until their discovery of HyperSpace, which was thought by Humans and most other Alliance races to be unique until recently. The English term "dimension" in this context was first coined by Androsynth scientists, by analogy with similarly named entities in Human speculative fiction, named as a shorthand for what are properly multiple sets of classical dimensions separate from the four TrueSpace dimensions we habitually move through. The Androsynth were the only TrueSpace race known to have done serious work studying dimensional physics; the Arilou have their own terminology and refer to the set of all dimensions and their interactions by a term akin to "time" in English, but when discussing these matters with the Alliance have mostly adopted Androsynth terminology.
Before the discovery of the Androsynth discipline of dimensional physics, a common metaphor for HyperSpace's relationship to TrueSpace was that of two planes parallel to each other in space, one "above" the other. This gives rise to the metaphor of "pushing up" into HyperSpace (itself a term meaning "above space") and "falling down" into TrueSpace, and the metaphor of dimensions as "higher dimensions" to TrueSpace. The Pkunk and some others commonly use the term "plane" for what we call dimensions, and the analogy of "upward" and "downward" appears to have been adopted by the Orz, referring to QuasiSpace as "above" and the Orz home dimension as "below", suggesting that their home dimension is a "lower dimension" that bears some analogous relation to TrueSpace as TrueSpace does to higher dimensions.