A Military Academy in the free game C-evo is a State Improvement that replaces the Barracks in one city. Instead of starting life "hardened" (i.e. with two stripes), as they do with a Barracks, ground units built in association with the Military Academy start life "elite", which is the top grade of strength enhancement (200% of the basic strength), indicated by a star where lower-grade units show stripes.
But there are costs: construction 100 resources (less the 40-gold sale price of the barracks); and whereas barracks maintenance is 1 gold per turn, the Academy costs 4 gold.
The effect on healing of damaged troops is the same as for barracks: health double each turn until they reach 100%.
There is no apparent relationship between a Military Academy and the Academy Training Unit Advance (available with Sun Tzu's War Academy).