I write this note because the subject matter as treated on the web is all rather advanced. I want to start at an intuitive level but hopefully advance to explaining some mathematics necessary to understanding General Relativity.
I hesitate in this effort as I remember a book I studied as a teen ager. I learned the meaning of all the sentences but still did not understand tensors. I began to understand tensors after reading a short monograph which I cannot now locate by Roger Penrose on the Christoffel symbol Γijk.
To locate a point on a manifold takes n real numbers, just as locating a point in 3 dimensions requires 3 numbers, or 2 reals (latitude and longitude) locate a point on the Earth. Navigators flying near the poles of the Earth sometimes use a nearly rectangular grid near the poles in preference to polar coordinates. There is a patch near each pole where the rectangular grid is used. Mathematicians do the same trick in reasoning about manifolds so as to avoid a great deal of extra reasoning about what happens just at the pole. They speak of an atlas which consists of a set of charts. Each chart is a bijective map between a subset of the manifold and a hypercube in Rn. Each point on a chart corresponds to just one point on the manifold and every point on the manifold is mapped by at least one chart. It appears at most once on a given chart. Even with the polar grid, however, the X and Y coordinates on the Earth below are not quite perpendicular, especially as you get farther form the pole. The Earth is not flat.
Each of these charts will be flat n dimensional maps of a curved portion of the manifold. A chart comes with a set of n coordinates. When we imagine these coordinates on the manifold, as carried over by some particular chart, they may be distorted, just as lines of longitude are closer near the poles. It is worse; they are not in general even perpendicular to each other.
G.B.H. Riemann worked much of this out in about 1850. They were built on earlier ideas from Gauss but Gauss stuck to 2D surfaces embedded in 3D. If you hold n−1 of the coordinates constant and change the other a small amount you move along the surface of the manifold a short distance. If you fix each coordinate except xi, and you increase xi by d then you get a vector d ei. As d goes to 0 ei tends towards a limit. It is a shaggy dog story as to why we use superscripts to name the coordinates. But it is a very good reason.
If the manifold were flat for this chart then we could arrange that the vectors ei could be perpendicular and of length one.
No such luck in general.
For some mathematical purposes we can get this far with no notion of distance between even nearby points on the manifold.
If we have such a notion then we collect for each combination of i and j the dot product of ei and ej and get the metric tensor
gij = ei • ej . gij will vary from point to point but it gives us the shape of this mapped portion of the manifold and along with such information from other maps, the shape of the entire manifold.
You might wonder what vector space this vector is in. Mathematicians speak of a tangent space, sort of as if it were a tangent to the manifold in some embedding space. If you study tensor bundles they make all this talk legitimate without finding an embedding space. We speak sloppily here as if the vector were in the manifold.
For the Earth the metric tensor (assuming a sphere) is