Q:

What does "&" mean in latex?

I am using the align command to align equations :
\begin{align*}
  \psi \colon & SU(2)\otimes SU(2) \otimes SU(2) \to  \mathbb{C}\\
  &(U_{l_1} \otimes U_{l_2} \otimes U_{l_3})  \mapsto  \psi(U_{l_1} \otimes U_{l_2} \otimes U_{l_3}).
\end{align*}

As I understand it, the "&" tells to latex where the object need to be vertically aligned.
But does & mean something in general in latex ? Or is it a command that has always different significations ?

A:

By default, & is exactly the alignment character, and even has its own category code (4). It is defined this way in both TeX and LaTeX.  For more on category codes see What are category codes?.
As with almost everything in TeX, it could in principle be reassigned to do something else, but usually it is not. TikZ in fact does reassign it, because in  TikZ matrix contexts it means \pgfmatrixnextcell, but this is a special case.
So for the most part, yes, & is the alignment character used in tabular material and other alignment structures like align and array, etc.  
For more discussion on why & is always special even outside of these environments see:

Why is & special outside of tabular environments?

