Prerequisites: Barracks Level 3, Main Building Level 3

In the academy new unit types can be developed. By increasing its level you can order the research of better units. Once a unit type is researched in a village it remains researched, even if the academy has been demolished. 
Some troops also require other buildings in order to be researched. For example, the Imperian (Romans) needs an academy level 5 and a smithy level 1. Look at the troops page to view the prerequisites.
The starting units (Phalanx, Clubswinger and Legionnaire) do not need to be researched in any of your villages and as such do not show up in the academy.

Development costs for Roman Troops
lvl
1		Praetorian	700	620	1480	580	1:58:00
5		Imperian	1000	740	1880	640	2:06:00
5		Equites Legati	940	740	360	400	1:38:00
5		Equites Imperatoris	3400	1860	2760	760	2:42:00
5		Equites Caesaris	3400	2660	6600	1240	3:26:00
10		Battering Ram	5500	1540	4200	580	4:20:00
15		Fire Catapult	5800	5500	5000	700	8:00:00
20		Senator	15880	13800	36400	22660	6:47:55

Development costs for Teutonic Troops
lvl
1		Spearman	970	380	880	400	1:26:00
3		Axeman	880	580	1560	580	1:30:00
1		Scout	1060	500	600	460	1:26:00
5		Paladin	2320	1180	2520	610	2:30:00
15		Teutonic Knight	2800	2160	4040	640	2:58:00
10		Ram	6100	1300	3000	580	4:00:00
15		Catapult	5500	4900	5000	520	8:00:00
20		Chief	18250	13500	20400	16480	5:23:45

Development costs for Gallic Troops
lvl
3		Swordsman	940	700	1680	520	1:42:00
5		Pathfinder	1120	700	360	400	1:38:00
5		Theutates Thunder	2200	1900	2040	520	2:34:00
5		Druidrider	2260	1420	2440	880	2:38:00
15		Haeduan	3100	2580	5600	1180	3:06:00
10		Ram	5800	2320	2840	610	4:40:00
15		Trebuchet	5860	5900	5240	700	8:00:00
20		Chieftain	15880	22900	25200	22660	6:47:55

The level dependent attributes (such as construction costs and generated culture points) can be found here.
For the full table of construction times click this link for server with 1x speed, this link for server with 2x speed and this link for server with 3x speed.