This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.
Generally, a tensor is an element of a tensor product.
Traditionally this is considered in differential geometry for the following case:
for $X$ a manifold, $T X$ the tangent bundle, $T^* X$ the cotangent bundle, $\Gamma(T X)$, $\Gamma(T^* X)$ their spaces of sections and $C(X)$ the associative algebra of functions on $X$, a rank-$(p,q)$ tensor or tensor field on $X$ is an element of the tensor product of modules over $C(X)$
A rank $(p,0)$-tensor is also called a covariant tensor and a rank $(0,q)$-tensor a contravariant tensor.
(…)
A vector field is a rank $(1,0)$-tensor field.
A Riemannian metric is a symmetric rank $(0,2)$-tensor.
A differential form of degree $n$ is a skew-symmetric rank $(0,n)$-tensor.
A Poisson tensor is a skew-symmetric tensor of rank $(2,0)$.
For instance section 2.4 of