A configuration space of a physical system gives you all the possible states it can be in. For example, if the system consists of an air molecule moving around in a room, then configuration space can consist of all the points in the room: that's all the positions the particle could be at, and it's a portion of ordinary 3D space as we know it. Each point in this configuration space can be pin-pointed by three coordinates $(x,y,z)$.

Where are all the air molecules?

If you have two air molecules then the configuration space consists of all the possible configurations of the two molecules, so it consists of pairs of points from 3D space. If particle 1 is at point $(x,y,z)$ and particle 2 is at point $(a,b,c)$, then that configuration corresponds to the point $(x,y,z,a,b,c)$ in configuration space. We can't visualise that space any longer because it's six-dimensional. That's not a problem though, as we know that it stands for two particles in ordinary 3D space.
If you have many air molecules in your room (which we hope you do, otherwise you'd be dead), the configuration space has lots and lots of dimensions: if there are $N$ particles, then configuration space has $3N$ dimensions, three for each particle.

You may also want to encode some information about the molecules' motion, for example their momentum, in configuration space. Momentum has three components, one for each direction in space. Therefore, if we include momentum, each particle comes with six pieces of information (three for position and three for momentum). The configuration space of a system of $N$ particles now has $6N$ dimensions.

This shows that configuration space, though based on our ordinary notion of 3D space, can be a lot more complicated than ordinary 3D space. The above all applies to classical physics as we learn it in high school. In quantum mechanics, however, things get a lot more complex and subtle. In this case configuration space can be even more exotic.