The term generally refers to theories of physics involving more than three dimensions of space and one of time. A large number of such theories exist and we could even say that most current activity in theoretical physics and cosmology involves hyperdimensional theories.
The drive to unify the four forces of electromagnetism, strong and weak nuclear forces and gravity has given rise to various models of the universe with more than 4 dimensions. The first widely known such attempt was Kaluza's and Klein's introduction of an additional looping space dimension for modeling electromagnetism.
Ever since the 1970's, string theories have operated with spaces of 11 or 22 dimensions, where the extra dimensions are typically circular and collapsed to sizes at the atom scale.
Various cosmologies involving the 3+1 dimensional spacetime of general relativity being embedded in a higher order universe exist. The M-brane theories for example see multiple 3+1 dimensional universes embedded in a 5 dimensional hyperspace, a bit like sheets of paper floating in space. These universes may sometimes come in contact with each other, giving rise to anomalous effects such as creation of matter from apparent void. The ekpyratic theory of cosmology is one such theory. Generally, gravity is the only force that bridges between these 'branes.' Some anomalies in the motion of spacecraft suggest the presence of invisible sources of gravity and indirectly could indicate a very weak interaction with other 'branes' or universes. These may variously consist of matter or antimatter. One explication for the relative weakness of gravity is that it would act along more dimensions than the 3 visible space dimensions.
General relativity is expressed in 3+1 dimensions but allows for anomalies in the structure of spacetime, such as direct connections between otherwise separate points and closed folds of space. At a point in the late 1930's, Einstein worked in a unified field theory involving a physically real 4th space dimension a la Kaluza-Klein. This work was not continued by Einstein, at least not in public, but the Cassiopaea material suggests this was completed by others and resulted in a unified field theory with technological applications.
The Cassiopaea material discusses hyperdimensional physics in the form of various clues but these are difficult to understand without much background. In general the material favors the existence of a 4th space dimension forming a double loop. This 4th dimension becomes perceptible from the 4th density onwards and allows a sort of simultaneous viewing of objects from the outside in as well as inside out.