I think of a dimension as something that I (or something) can move through.
Well then in that case there are 4 dimensions you can move through and actually notice that you are moving through them.
1. Width
2 Length
3. Height
4. Time
...again the labels "width", "length" and "height" are not quite so accurate in describing the first three dimensions, because the 3 mix and merge together to form 3-dimensional Space. The 4th dimension Time is distinct. Its not worth talking about the 5th and further dimensions without twisting up one's brain into all kinds of knots, but the crux of the matter is that some particles are able to escape into the further dimensions thereby weakening their force or pull of attraction in the 4 dimensions that we can sence. Gravity for example is a very weak force in relation to some of the others, so it is said to exist primarily in a further dimension.
Physicists have seen particles as disipating energy in other dimensions by positing them instead of non-dimensional points, as multi-dimentional strings, or multi dimentional (mem)branes. The singularity which started the Big Bang is said to have been an infinately small point with no length width breadth or time. It then unravelled into more and more dimentions. Think of this...
a dot/period is 0-dimensional
a line drawn on paper (string) is 1-dimensional, it is "long" and it has "length"
the piece of paper (membrane) itself is 2-dimensional
and a whole wad of paper is 3-dimensional
the wad of paper is 4-dimensional... due to the fact that it doesn't just POP into existence for one micro-second then disappear from existence... it exists for a long "Time"
This view is very simplistic because even the dot/period has 4 dimensions when you go onto a smaller level of scale, but these examples are just to help one get ones mind around the concept of dimensions. When we look at the edge of a pice of paper, it seems like a 2-D line, when in fact there is a whole piece of paper just beyond our vision... Similarly we are mutli-demensional due to the fact that many of the particles that make us up seem to be specks, when in fact they exist in other dimensions as strings or membranes that are "beyond our vision" as it were... the "dot" that is the particle, is only part of its protrusion into our level of dimensional awareness... Note too that linguistically, we use the word "length" to describe both physical and spacial length, and a length of time.