I've nothing to add to Catholic Scientist's informative post. I just wish to explain the term manifold, in case the Wikipedia page is not sufficient.

Essentially a manifold is an shape where you can label the points using a set of real numbers. The number of real numbers needed is called the dimension of a manifold.

For example the Earth's surface would be a two-dimensional manifold as any point on its surface can be labelled using two real numbers (longitude and latitude).

The space of all states of a classical particle is also a manifold, any state can be described using six real numbers (six-dimensional manifold), three for the position of the particle and another three for the value of its momenta in each direction.

Some shapes (or spaces, which is the technical term) are too bizarre to be described using real numbers, although the use of these spaces is limited in physics.

If the manifold has a notion of distance between its points, then it is called a Riemannian manifold or a Pseudo-Riemannian manifold if one of the dimensions is time. The surface of the Earth is a Riemannian manifold, since there is a distance between points on the Earth's surface. The space of states is not, as you can't really say how far states are from each other.

How the distances work on a particular manifold is given by an object called the metric. Einstein's Theory of General Relativity basically says that the energy density of matter determines the metric of spacetime.

That is, the energy density of matter at a point affects how the rules of distance work near that point.