It is called Metric tensor to a 2-covariant tensor or type (0.2) whose coefficients form a nonsingular symmetric matrix
An important example of the metric tensor is the usual scalar product in the plane
in fact, gij is a tensor of type (0, 2) and applying two vectors in R2 in real numbers.
G(v,w) = vigijwj
The 'inverse tensor' of the metric tensor is the contravariant metric tensorgij
Givcen a tensor Tj (1,0), the contrast with the metric tensor gij, poduces a (0,1) tensor. sometimes it is called the Tj contravariant coordinates of the tensor Ti and covariant coordinates
Note: Obviously, if a tensor is a metric tensor in a basis, it will be in all.