Don't feel to bad about being confused as there isn't a definitive answer to your question.

In short though, there is infinite n-1 dimensional space within an n-dimensional environment. What this means is that, for example, there is room for infinite area, which is two-dimensional, within a volume, which is three-dimensional, or that there is space for infinite length i.e. one-dimensional, within a two-dimensional area.

Probably the best illustration of this is a plot of the Mandlebrot set fractal:

http://en.wikipedia.org/wiki/File:Mandelset_hires.pngWhen you plot the Mandlebrot set you get the fractal picture shown in that link above, where the 'black' area indicates the combinations of x and y values that don't resolve, and which therefore lie in the 'set', and where the white area surrounding the black area indicates where the number pairs do resolve, and which are therefore not members of the 'set'. As you can see, the area of the set i.e. the black area, is not infinite, but if you try to measure the length of the perimeter between the two areas you'll find that it is infinite. We thus seem to have something that has finite area but with an infinite length perimeter (in the more colourful plots of the Mandlebrot set, the different colours just represent how many iterations are required to resolve the value pairs that fall outside the set).

See

http://en.wikipedia.org/wiki/Mandlebrot_set for the full article.