The

metric tensor is more precisely a

symmetric bilinear form which gives rise to a

Riemannian metric. To clarify, you can write is as a symmetric

matrix A

_{ij}, and then write the metric in the form

ds^{2} = A_{ij}dx_{i}dx_{j}

where x is the coordinate on your manifold. Note that because of the symmetry of A, it will have 3 independent components in 2-d, and 10 independent components in 3-d. The above reduces to the Euclidean metric when A is the identity matrix, and then

ds^{2} = dx^{2} + dy^{2} + dz^{2}

which is a

differential statement of

Pythagoras's Theorem.

In G.F.B. Riemann's scheme of geometry, the metric tensor must be positive definite, that is to say that A has strictly positive eigenvalues, in order that all distances are measured as being positive. However, in the Special and General Theories of Relativity, 3 of the eigenvalues are positive and 1 is negative, and so it these cases, the metric is said to be pseudo-Riemannian (it had the effect that distances measured inside a light cone all turn out to be zero). If you think about it, it is precisely this condition which makes space and time different (read more about it here).