What does the symbol on Edward's coat mean?
It's called the Flamel (yes, named after Nicolas Flamel, purported to be an alchemist who sought the Philosopher's Stone).

It is most noticeably similar to the symbol of Hermes, a god of alchemy, and also the Rod of Asclepius, which you may recognize from ambulances as the EMS symbol.

 

It is not a religious symbol, nor is it ever explained to have a relevance in the FMA universe. However, given its real life links to alchemy and medicine, it's safe to say that the author intended to bring forth real-world alchemical symbols into her universe.

Further reading: FMA Wiki entry on the Flamel