Numbers are basically a way for people to organize their thoughts about space and time. First we can think of zero as a point, taking up no space at all. Then we begin to count by using numbers to count how many people there are, or how many sheep we have. That's the positive half of the number line, going off past billions and trillions to infinity.
But soon we realize that we need to imagine a negative half of the number line too - we might have less than no sheep, if we were in debt. We can also use this number line as a time line, showing which things happened earlier and which things happened later. Check out a Christian time line here. The negative number line also goes off to infinity.
We also realize that there are smaller numbers in between the whole numbers you see marked on the line: we can have one and a half acres of land, or two and a quarter acres. We call these smaller numbers fractions.
Now we think about it a little more, and we realize that we can use numbers to show how big our room is - the area of our room. To do that, we'll need to write down the width and the length of the room. That's two numbers that we're going to multiply together to get the area. We can think of these numbers as covering a plane - a flat sheet of numbers stretching out to infinity in every direction. This is often represented by showing numbers along an x and a y axis.
Now, we realize that if we live in a three-dimensional world of solid objects and not just flat ones, then numbers may exist in a solid world too. Perhaps we ought to think of numbers along an x axis, a y axis, and a z axis. That way we could use numbers to describe a solid object like a house or a chair - that would be very useful. We could combine numbers with geometry to make the kind of mathematics we call algebra.
Finally, we know that we live not just in a three-dimensional world, but also in a universe that measures time - a sort of fourth dimension. It's hard to draw it, but we can imagine numbers that go off from the center x,y point in four dimensions or even more. There could be an infinite number of number lines, in an infinite number of dimensions.