@Puzzledandpissedoff - a line is a one-dimensional manifold. "One dimension" just means that there's only one direction that you can move without leaving it - forwards-or-backwards. One side of a piece of paper is a two-dimensional manifold. This time there are two directions you can move, forwards-or-backwards and left-or-right. You can move in other directions, but they're all a combination of moving a bit forwards or backwards at the same time as a bit left or right. And space is an example of a three-dimensional manifold, because you can also move in a third direction, up-or-down.
Mathematicians love to complicate things, though, and say "why stop at three directions just because nature does? Can we think about entirely made up manifolds where you can move in four or five or a hundred different directions?" That's where the "arbitrary dimensions" comes in. I'm not sure if Riemann was actually the person who worked out all the maths of these things, but he had a lot to do with it and a whole bunch of them are named after him.
Anyway, then Minkowski (a professor who'd tried to teach an uninterested student by the name of Einstein about these imaginary mathematical things) pointed out that one of them, a four dimensional one where one direction is a bit different from the other three, is a really handy thing for understanding Einstein's theories. Einstein took that idea and ran with it, developing his Special Relativity into General Relativity, which is the theory that predicted the gravitational waves we've recently detected. So now we think that space is only a part of a four-dimensional structure we call spacetime and we try not to be rude about mathematicians over-complicating things. Because it turned out they weren't.
I hopethat makes some kind of sense. 
It's possibly more than you wanted to know on a random Tuesday...