Not sure if all the math gurus here agree, but let me give you my interpretation.
I think to understand what a dimension is, you must understand the concept of space first.
There is physical space - the one you know about. A dimension is some direction in the space: so there's three spatial dimensions, and maybe time, right?
So the minimal number of coordinates you need to specify a unique point in the physical space is 3 or 4 (depending on if you count time) - ironically this number is also called the dimension of space.
But there might be also other -more abstract- spaces! For instance, you can keep track of the amount of apples you have, the amount of banana's and the amount of pears! Then, in this space of fruit you can have the point (1,2,3) referring to the fact that you own 1 apple, 2 banana's and 3 pears. The dimension of this space is three, and the number of dimensions in it is also three (as you see the language is pretty poorly designed here). Of course by adding pineapples and grapefruits you can easily go to 5 dimensional space.
This is of course not a space as in the conventional sense of the word, but more a mathematical sense: space and dimensions are just a concept to keep track of things.
Most of the time spaces have some added structure, like some concept of 'distance', but as can be seen from my fruity example, sometimes that's not even the case! For instance is the point where you have 1 apple closer of further from the point where you have only one pear than from the point where you have only 1 banana? But that may be too far from you question.
