What does the symbol on the back of Edward Elric's jacket mean?

It is based on Kabbalist and Hermetic alchemy symbolism. It has two major elements, the Winged Crown and the Crucified Serpent

The winged crown symbolizes the mastery or achievement of magisterium over the volatile or some sort of alchemical perfection. The crucified serpent means the stabilization, manifestation or incarnation of the volatile.

Volatility is based on the idea that all things, or material, is made of fire or energy. To stabilize and manifest fire is to give it a form in the physical, or material sphere.

The symbol, which is shared by Edward, his brother Alphonse, and their teacher Izumi Curtis, is an old symbol that represents alchemy itself.