The person to give you a proper answer is
@Polymath257 for its use in geometry and mathematics more generally. But just until he shows up.....
.....a point in physical space can be defined in terms of values along 3 spatial dimensions, i.e. 3 independent axes of measurement. Time is a fourth, non-spatial dimension. Famously, relativity uses a 4dimensional conception of physics in which the time dimension and the 3 spatial ones are put on the same footing. There are some speculative models that try to combine quantum theory and relativity (I think) by defining more dimensions. The essence of a dimension in this sense is that it is independent of all the others, at right angles to them all, effectively. In mathematics, there is nothing to stop you defining as many of these as you like, algebraically, even though it is not possible to picture them physically because physical space has only 3.
In science generally, a dimension is one of the fundamental measurable quantities, including mass, length, time and electric charge, in terms of which all other measurable quantities can be expressed. For example, energy has the dimensions: mass x length squared/time squared, normally written ML²/T². Force has the dimensions ML/T². (Since mechanical work, a form of energy, is force x distance, you can see how this checks out.) Checking the dimensions on both sides of an equation is a good way to verify that you have not gone off the track somewhere in your analysis.
In sci-fi, people have a tendency to use the mathematical idea of unlimited dimensions to describe alternative universes as being "in another dimension", or some such. This does not really make much sense, since any non-trivial universe, i.e. one with actual objects and actual physics in it, would need several dimensions to measure what was going on in it.
But I may be describing this badly, so I'll let others have a go.