A matrix is diagonalizable, so what? I mean, you can say it's similar to a diagonal matrix, it has $n$ independent eigenvectors, etc., but what's the big deal of having diagonalizability? Can I solidly perceive the differences between two linear transformation one of which is diagonalizable and the other is not, either by visualization or figurative description?
For example, invertibility can be perceived. Because non-invertible transformation must compress the space in one or more certain direction to $0$. Like crashing a space flat.

Up to change in basis, there are only 2 things a matrix can do. 


*

*It can act like a scaling operator where it takes certain key vectors (eigenvectors) and scales them, or 

*it can act as a shift operator where it takes a first vector, sends it to a second vector, the second vector to a third vector, and so forth, then sends the last vector in a group to zero.


It may be that for some collection of vectors it does scaling whereas for others it does shifting, or it can also do linear combinations of these actions (block scaling and shifting simultaneously). For example, the matrix
$$ P
\begin{bmatrix}
4 &  &  & &  \\
 & 3 & 1 & & \\
 &  & 3 & 1 &\\
 & & & 3 & \\
 & & & & 2
\end{bmatrix} 
P^{-1}
=
P\left(
\begin{bmatrix}
4 &  &  & &  \\
 & 3 &  & & \\
 &  & 3 &  &\\
 & & & 3 & \\
 & & & & 2
\end{bmatrix}
+
\begin{bmatrix}
 0&  &  & &  \\
 &  0& 1 & & \\
 &  &  0& 1 &\\
 & & &  0& \\
 & & & &0 
\end{bmatrix}\right)P^{-1}
$$
acts as the combination of a scaling operator on all the columns of $P$ 


*

*$p_1 \rightarrow 4 p_1$, $p_2 \rightarrow 3 p_2$, ..., $p_5 \rightarrow 2 p_5$,
plus a shifting operator on  the 2nd, 3rd and 4th columns of $P$:

*$p_4 \rightarrow p_3 \rightarrow p_2 \rightarrow 0$.
This idea is the main content behind the Jordan normal form.
Being diagonalizable means that it does not do any of the shifting, and only does scaling.
For a more thorough explanation, see this excellent blog post by Terry Tao:
http://terrytao.wordpress.com/2007/10/12/the-jordan-normal-form-and-the-euclidean-algorithm/
