There are three main branches of Galifrey's government. The first is the People's Council, a group of one hundred and one male citizens elected by popular vote. The second branch is the High Council, which is a group of nine male aristocrats appointed by the Elder's Council, a group of all male citizens over the age of sixty.
The High Council does the majority of the governing, being smaller and able to meet most frequently. Their primary responsibilities is managing the treasury, trading and diplomacy with other city-states, and passing legislation. The nine members also serve as generals and admirals in times of war.
The People's Council meets much less frequently and generally only votes on issues such as declaring war. They are also in charge of collecting taxes and distributing food.
The Elder's Council is considered the most powerful of the three councils. They serve as judges and jury and lawyers for legal cases. They have little to do with legislating, however, they have veto power on any bill or action which the People's Council or the High Council attempt to enact.