The Academy is very important. Your academy research can speed the production time of buildings, troops, or the gathering of resources. In the later stages your academy can provide bonuses to your troops in battle or longer range for your archers. An academy is one of the first buildings I create in a new city.
When doing research in the academy the hero you have as mayor can help speed up the time it takes to research. If you have a mayor with high intelligence when you start the research project the amount of time it takes to perform the research will be speed up. As soon as you begin the research you can go back to your regular mayor with high politics.