In mathematics, we can speak of a space with a certain number of dimensions. A sheet of paper has 2 dimensions, i.e. two coordinates are needed to define the position of a point on it. Three coordinates are needed to define the position of a point inside a solid.
A dimension is not simply an axis perpendicular to all other axis of the space in question. Dimensions can be linear or cyclic, finite or infinite. The sheet has two finite dimensions. Rolling the sheet into a cylinder continues to have two dimensions but now one of the dimensions is cyclic. Such different configurations of a space with a certain number of dimensions are called manifolds.
In popular parlance, the word dimension is often used in the sense of a parallel or alternative reality.
The Cassiopaea material contains both uses of the word.
The term density is different from dimension. Density refers to an entity's degree of development and to a mode of perceiving and interacting with the world that corresponds to this level. It is quite possible that entities of a higher density perceive more dimensions than the three dimensions of space and one of time that the human form is naturally comfortable with. Still, a density is not a dimension.
Often in New Age literature the concepts of dimension and sometimes of density or plane of existence are used without much discipline, often interchangeably. For example the phrase 3rd dimensional life form would refer to 3rd density life form in the Cassiopaea nomenclature.