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 Aij, and then write the metric in the form
ds2 = Aijdxidxj
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
ds2 = dx2 + dy2 + dz2
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).