The Galactic Terran Alliance is just that: an alliance of Terran worlds. They've given up their control of external policy in exchange for military protection. Each individual world, or group of worlds, retains its own government and has internal autonomy, but the GTA has control when it comes to matters involving two or more member worlds.
Exact governmental mechanisms probably consist of a council to which each world/group of worlds elects, or appoints, a member. The GTA also represents the only significant Terran military force. It will protect its members, so they have no need of their own militaries.