Your military buildings are where you train your glorious army.
 

Infantry Camp - The Infantry Camp is the first military building unlocked. It has a fixed location in your city below the War Council. This building is for training Light Infantry units.


Barracks - The Barracks allows you to train Heavy Infantry Units like Peltasts.


Armory - The Armory allows you to train Phalanx units like the Thureophoros.


Stables - You actually unlock Stables very early on. However, you will only be able to train scouts with it until you unlock the Mounted Peltast. Since scouts are your means of scouting the enemy, they are important to a player's strategy and once you unlock other Cavalry units, you'll have to balance building strong offense/defense units vs. more Scouts.


Hall of Heroes - The Hall is unique among the Military Buildings in that it does not have a queue and only trains one unit, the Hero. The Hero is a versatile unit that costs no resources. You can only train so many each day though, and this is done via the Hall of Heroes.



Additional Notes: As with most buildings, military buildings are unlocked through the agreements. To train a unit though you must unlock both the building and then unit as well as have the building built in your city. Each building maintains it's own training queue allowing you to train one unit from each tier at any given time. You can queue up multiple units for each tier but understand that once you have selected to train a unit, it can't be cancelled and you can't alter a production queue already in progress. Thus, if you accidentally queue up 5 days worth of Hoplite, you'll have to either wait 5 days or boost production before you can train different light infantry units. Since boosting is expensive and there are tons of other uses for your Drachmas, you should queue up units carefully.