If you have any linear algebra background, then the definition of a tensor is straightforward: given a vector space V over a field K (in physics, K = R or C), a tensor T is a multilinear (i.e. linear in each argument) function from vectors and dual vectors in V to numbers in K. That's it! A type (p, q) tensor T takes p vectors and q dual vectors as arguments (p+q is often called the rank of T but is ambiguous compared to the type).
(If you're unfamiliar with the definition of dual vector, it's even simpler: it's just a linear function from V to K.)
(If you're unfamiliar with the definition of dual vector, it's even simpler: it's just a linear function from V to K.)