What is the meaning of Cotangent bundle tensor product tangent bundle:
$T^*M\otimes TM$? what will an element of this space be?
This is isomorphic to $Hom(TM,TM)$. Namely it is the endomorphism bundle of the tangent bundle. In general given two vector spaces(or bundles) we have an isomorphism between $W \otimes V^*$ and $Hom(V,W)$.