Significance and physical meaning of diagonalization of linear maps and bilinear forms, eigenvalues and eigenvectors In linear algebra, I have studied the diagonalization of a linear map and of a bilinear form; and also the concepts of eigenvalues and eigenvectors. 
However, the importance of diagonalizing a linear map or a bilinear form and the significance (and physical meaning) of eigenvalues and eigenvectors has never been properly explained to me. 
Could you clarify these points (also by pointing out some references)?
 A: As a general reference may I suggest Wikipedia... also I am not going to address bilinear forms.
For the question about eigenvalues and eigenvectors please see What is the importance of eigenvalues/eigenvectors?
If a linear map on an $n$-dimensional vector space $T:V\rightarrow V$ is diagonalised then it may be represented as a diagonal matrix. A diagonal matrix is of the form
$$T=\left(\begin{array}{cccc}\lambda_1 & 0 & \cdots & 0
\\0 & \lambda_2 & \cdots &0
\\ \vdots & \vdots & \ddots & \vdots
\\ 0 & 0 & \cdots & \lambda_n \end{array}\right).$$
If you know how a matrix represents a linear map then you will know that this implies that we have written $T$ with respect to an ordered basis of $V$, $\mathcal{B}=\{e_1,e_2,\dots,e_n\}$, such that each of the elements of $\mathcal{B}$ are eigenvectors of $T$:
$$Te_i=\lambda_i e_i.$$
A little more can be said but this is a good initial foray.
A: The best physical intuitive and interactive explanation of eigenvalues/eigenvectors can be found in the link below
http://setosa.io/ev/eigenvectors-and-eigenvalues/
