Intuitive

A tensor is a relation between one vector and another. If you start with one vector, such as a force, and mathematically apply it to a tensor, then you get another vector. That vector might be, for example, the stress caused by the force.

That’s the simplest thing they do. Tensors do other things too; for example, the metric tensor represent the geometry of space. A tensor can represent the energy-momentum density. A tensor can represent a combination of electric and magnetic fields in a way that some of Maxwell’s equations greatly simplify. But the simplest and most basic connection is the vector to vector one.

A function relates one number (a “scalar”) to another one. A tensor does that for vectors.http://qr.ae/TUTNzc

Concrete

Recommended Books

• A Student's Guide to Vectors and Tensors by Fleisch - a nice student-friendly introduction
• Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters (free)
• A Brief on Tensor Analysis by James Simmonds - a concise but great introductory text.

Abstract

Why is it interesting?

Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalization of classical linear algebra. In classical linear algebra one deals with vectors and matrices. Tensors are generalizations of vectors and matrices. Introduction to Tensor Calculus by Kees Dullemond & Kasper Peeters