Introduction
A given object or entity is referred to as being higher-dimensional when they exist as part of a system with a number of coordinates axes greater than our own, or in layman's terms, if they possess more than three dimensions.
Speaking in simple terms, a "dimension" can be considered an axis of movement and measurement, through which beings and objects can displace themselves throughout a given physical system. 1-dimensional beings would only be able to dislocate themselves to left and right across a single direction, while 2-dimensional beings would have an additional spatial axis inherent to them, and thus be able to move up and down as well. Finally, 3-dimensional beings like ourselves are able to displace themselves left and right and up and down, as well as forwards and backwards, encompassings parts of additional axes of space.
Following this same principle, a 4-dimensional being would be capable of moving through an additional, perpendicular direction incomprehensible to lower-dimensional beings, and this can be generalized unto any higher number of dimensions.
Mathematical Intuition
Speaking in more rigorous terms, a "dimension" in a system can be most effectively equated to the real number line, which is itself the most basic example of a 1-dimensional coordinate space (commonly denoted as simply as ℝ), and contains bounded subsets (smaller, finite lines if you will) whose exact measurements can be taken by "pointing" at a real number represented as a specific point in the line (Example: A line with length 3 is a subset of the real number line, whose length is infinite)
In this context, adding another dimension to a given space is effectively adding another real number line in a different, perpendicular position, which is mathematically represented by continuous cartesian products of ℝ. For example, 2-dimensional space is represented by ℝ^2 (ℝ x ℝ), 3-dimensional space by ℝ^3, and so on and so forth.
In this context, measurements of positions in space are once again obtained by directing oneself at arbitrary numbers contained in the axes of the given expanse, thus obtaining one's own coordinates in it. Simplifying this case, one could also informally state that the "Dimension" of a space is the exact number of coordinates needed to specify any position within it.
Likewise, any n-dimensional object can be thought of as being a subset of a real coordinate space of corresponding dimension. Here is a good illustrative example of that.
Take note that, while higher-dimensional objects can indeed be mathematically modelled and defined as uncountably infinitely larger than lower-dimensional correspondents, such abstract implications are not necessarily universal and applicable under any framework, and given the vast range of interpretations regarding higher-dimensional beings present throughout fiction, we have choosen to default to a more physics-based framework instead, and thus, being higher-dimensional does not necessarily skyrocket a character to a higher tier.
However, higher-dimensional characters can indeed be ranked at higher tiers if:
A) They are provably infinite on all axes, or can safely assumed to be so.
B) Are defined as always being infinitely larger or superior to lower-dimensional beings by the fictional work which they are from
C) They correspond to large extra dimensional spaces, such as those defined in brane cosmologies and the like.
Applications
Keep in mind that this is merely a very general list of the abilities that a higher-dimensional entity would possess, and is not necessarily applicable to all of fiction.
Pseudo-Invulnerability: Due to the additional axis spatially inherent to them, higher-dimensional objects are comprised of more information than what can normally be described in lower-dimensional spaces, and thus may only be capable of interacting with them through lower-dimensional "slices" or cross-sections of their bodies.
Pseudo-Teleportation: Higher-dimensional entities may be capable of seemingly phasing in and out of sight by moving through additional axes imperceptible to lower-dimensional beings, granting them a notable advantage in overall range and mobility.
Unconventional Physiology: Although much of the same physical laws and principles would theoretically apply to higher-dimensional worlds, they would likely be instantiated very differently, resulting in matter behaving in strange ways compared to our own Universe.
Large Size: Due to the additional angles possessed by them, higher-dimensional objects and beings would be much larger than lower ones.