How to make a 6 x 6 board using the swing library, in netbeans, and with what elements is the best way to do it?

Use a GridLayout (Grid distribution), and in each cell add a JPanel to which you can put a background color and border if you need to distinguish some cells from others.

Here is a bit of code:

///Define el tamaño de la cuadricula. 

Nombre_De_Tu_Frame.setLayout(new GridLayout(6,6));

///for para agregar paneles.
for(int i=0;i<6*6;i++){
JPanel panel=new JPanel();
panel.setBackground(Color.red);  ///se asigna un color.
Border borde;
borde = BorderFactory.createLineBorder(Color.black);  ///se le pone un borde.
panel.setBorder(borde);
Nombre_De_Tu_Frame.add(panel); ///Se agrega el panel a la cuadricula una vez que tiene color y borde.
}