Interpretations of (r,s) tensors A tensor of type (r,s) on a vector space V is a C-valued function T on V×V×...×V×W×W×...×W (there are r V's and s W's in which W is dual space of V) which is linear in each argument. We take (0, 0) tensors to be scalars, as a matter of convention. The interpretations of (r,0) tensors are trivial, since they are definitions of multilinear functionals (as a special case (1,0) tensor interpreted as covector (elements of dual space)). We can interpret (1,1) tensors as follows: A(v,f ) ≡ f (Av). Say we have a linear operator R; then we can turn R into a second rank tensor T by T(v,w) ≡ v · Rw where · denotes the usual dot product of vectors. If we compute the components of T we find that the components of the tensor T are the same as the components of the linear operator R. Ok. Everything is good. But I cant understand interpretations of other (r,s) tensors. For example I found in Wikipedia (0,1) tensor intepreted as a vector or (0,2) as a bivector and in general (0,s) tensor as n-vector tensor; or (2,1) tensor as cross product and so on. I want you to show how the tensors in general interpreted. Is it possible for you to show these interpretations like as I did for (1,1) tensor ? Please dont close this post. My answer is not in "What is tensor ?"
 A: Just like $(r,0)$ are "multilinear functionals" assigning a group of $r$ co-vectors i.e. $r$ $(0,1)$ tensors a scalar, and you seem to accept this visualization, the $(r,s)$ tensors are "multilinear functionals" assigning a group of $r$ co-vectors i.e. $r$ $(0,1)$ tensors and $s$ vectors i.e. $(1,0)$ tensors a scalar.
This is just one way among many how to present or visualize given tensors. It is a way that suggests that one may "contract" all the tensor's indices with vectors to get a scalar, i.e. a $(0,0)$ tensor, as a result. But there are many other ways how the indices of tensors may be contracted to get other tensors (including scalars, vectors, covectors, and other tensors). You presented one such interpretation of the $(1,1)$ tensors – they're operators, either on the space of vectors or co-vectors.
Quite generally, the most general product of $(r_i,s_i)$ tensors for $i=1,2,\dots N$ without any contraction of indices is a tensor of the type
$$ (\sum_{i=1}^N r_i, \sum_{i=1}^N s_i) $$
which in plain English means that the number of lower and upper (covariant and contravariant indices) is simply being added from the individual products.
Concerning the elementary non-trivial (non-scalar) tensors, $(1,0)$ tensors and $(0,1)$ tensors are vector spaces that are "dual" (a couple of a sort) to each other, in the sense of linear algebra. One of these spaces contains all the linear forms acting on the other space, and vice versa. The relationship is symmetric, therefore it is a "duality". So if you think that you know an interpretation of one of these tensors, you should admit that you know an interpretation of the other space (and its elements, tensors), too.
Once you have "interpretations" for the elementary $(1,0)$  and $(0,1)$ tensors, you may have an "interpretation" for the most general $(r,s)$ one as a "multilinear functional" acting on $r$ vectors of one kind and $s$ vectors of the dual type.
