Here I will be just posting a simple questions. I know about vectors but now I want to know about tensors. In a physics class I was told that scalars are tensors of rank 0 and vectors are tensors of rank 1. Now what will be a tensor of rank $2,3\ldots$? This is quite tempting. So my question is: What are the prerequisites I need to learn profoundly before taking up an introductory course on "Tensors"?
I would like to suggest a quick shortcut:
Quick Introduction to Tensor Analysis, by Ruslan Sharipov.
This little pdf is self-contained, so you will need no prerequisites to read it. I believe it may well suit your need.
There are many good books on this subject. If you are comfortable with abstract setting and if you have taken a course in Linear Algebra then there is this book
"Tensors: The Mathematics of Relativity Theory and Continuum Mechanics" by Anadijiban Das. If you take any good book on relativity( for example "Landau's Classical theory of fields") you can find a sufficient enough introduction to tensors. You may also want to refer to the book "Differential Geometry" by Willmore where a nice introduction to Tensors was presented in Part2 of the book. All you have to know is basic Linear Algebra.(I am assuming that you have taken one or two courses in Basic Calculus)
I think you need basics of advanced algebra. I like the lecture notes in http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf . If you can't understand something in it, find details in some algebra book.
See also An Introduction to Tensors