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|
|Development costs for Teutonic Troops|
|Development costs for Gallic Troops|
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 and this link for server with 3x speed.