Dan Fleisch is a Professor in the Department of Physics at Wittenberg University. Here he briefly and very clearly explains vectors, vector components, and tensors.
YOUTUBE f5liqUk0ZTw What's a Tensor? (12min)
You may be wondering what is it about the combination of components and basis vectors that makes tensors so powerful.
All observers, in all reference frames, agree: not on the basis vectors; not on the compliments; but on the combination of components and basis vectors.
The basis vectors transform one way between reference frames, and the components transform in just such a way so as to keep the combination of components and basis vectors the same for all observers.
It was this characteristic of tensors that caused Lillian Lieber to call tensors "the facts of the universe".
A Student's Guide to Vectors and Tensors book 
The conventional explanations of tensors are harder to understand. wikipedia