The red cross is the coat-of-arms of the city of Milan and related to the Crusades, hence the cross. The figure being eaten by a crowned serpent is a child or a Saracen, depending on who you listen to.

Two symbols of Milan: the snake of the Visconti family on a light blue background, and a red cross on a white background, surrounded by a small metallic circle with the words ALFA-MILANO.

A.L.F.A. comes from the Italian Anomina Lombarda Fabrica Automobili (Lombarda car making company) and the original badge has Alfa written over the top of the badge and Milano underneath separated by two figure of eight knots. In 1920, five years after the take over by Nicola Romeo in December 1915 the badge changed to read Alfa Romeo above with Milano across the bottom. In 1925 a wreath was added to celebrate the marque’s numerous victorys on road and track, and this has since evolved into the gold trim which presently encompasses the badge. It was not until 1972 that Milano was dropped with the opening of the factory in Pomigliano d’Arco.

Useful links:

Official Alfa Romeo website (Italian)
Alfa Romeo wallpapers on Wallpapers.Lv
Alfa Romeo on Wikipedia