That's an excellent answer.  Of course Greek Fire is probably the most famous of their innovations, although the exact formulation(s) has been lost through the years.  The development of the phalanx as a formation in combat maximized the advantages of their armor, helmets, and shields, and worked well in close combat with the spear (the primary offensive weapon) and the straight and curved short swords, the xiphos and makhaira.  The blades of their weapons, of course, made from bronze were prone to bend, and quite different from the iron weapons of later times and those iron weapons used earlier by such cultures as the Hittites, and others in the areas of Palestine and the Levant.

Interestingly, some of the most innovative of Greek developments were in the areas of tactics and strategy, especially in their uses in naval warfare.  The naval victory at Salamis was perhaps the best example.  I suggest study of The Peloponessian War, an eyewitness account of the first "great war" of history, the 23-year struggle between the Athenian and Spartan alliances, written by Thucydides, a general during the war.

The Ancient Greeks were known as a strong and powerful society.  The Early Greek society was always at war with someone.  They fought with Rome, Persia, city-states within the country of Greece and Macedonia.  Their constant state of war was the catalyst for their inventions of military armaments.  The Greeks had the best foot soldiers in the world and is on what our infantry today is based.  The Greek soldiers were known as “hoplites.” 

The hoplites were supplied with armor which consisted of a shield to protect his body from spears, arrows and knives.  They also used their shields to break the weapons of other soldiers.  They also carried spears, double-edged swords made of iron, wood, and bronze.

Other weapons used by the Greeks and Spartans were the ballista from which several arrows could be shot at one time; a dagger, also called a short sword; a catapult which was used as artillery to throw large objects and stones at the opposing army.