Given the code :

 A = [1 2 3; 3 2 1]

 B = A.^2

The output :

B =

     1     4     9

     9     4     1

But if I do this : B = A^2

The output is :

Error using  ^

Inputs must be a scalar and a square matrix.

To compute elementwise POWER, use POWER (.^) instead.

What does the operator . do exactly ?

The dot itself is not an operator, .^ is.

The .^ is a pointwise¹ (i.e. element-wise) power, as .* is the
pointwise product.

.^ Array power. A.^B is the matrix with elements
A(i,j) to the B(i,j) power. A and B must have the same size, unless one of them is a scalar.

C.f.

• "Matrix and array arithmetic": http://www.mathworks.de/help/techdoc/ref/arithmeticoperators.html
• "Pointwise": http://en.wikipedia.org/wiki/Pointwise
• "Element-Wise Operations": http://www.glue.umd.edu/afs/glue.umd.edu/system/info/olh/Numerical/Matlab_Matrix_Manipulation_Software/Matrix_Vector_Operations/elementwise

¹) Hence the dot.