The Palace is where your national leader (or what passes for it) resides. Aside from training a few elite units and being where your patriots spawn, the Palace is also where Reforms and Policies are enacted.
The Palace is enabled by researching a despotic type of government (namely, Despotism), the Meeting House is then upgraded into the Palace. Aside from training elite units, it also has extra health, it can attack enemies and it can also detect cloaked units. Additionally, it provides a small wonder point bonus, thus allowing factions with Despotism a slight advantage in achieving Wonder victories.