Force sensitivity is apparently something one can be tested for. Babies born in the Republic are tested at birth, and those that pass a threshold are given to the Jedi Order. The Sith induct their recruits later, in the ten-to-fifteen years old range. Joining up does not seem to be entirely voluntary, in either case.
The Galactic Republic is a kind of federated representative democracy. Members are largely free to self-govern on the level of planetary systems, as long as those governments meet basic standards defined in the Republic's constitution. System governments send representatives to the Senate, which is the legislative body for the Republic as a whole. There are a few multiple-system subpolities, but these are rare, as they still only get the standard Senate representatives. The Senate elects from its members a Supreme Chancellor, who leads the Senate and wields executive power, though much of that is semi-permanently delegated to smaller committees within the Senate.
The Sith Empire is ruled absolutely by the Emperor. Theoretically. He hasn't given a direct command since the Treaty of Coruscant. In practice, the Dark Council rules the Empire, and is made up of the twelve most powerful Sith in the Empire. Each is responsible for a certain aspect of the Empire: Military Offense, Military Defense, Military Strategy, Production and Logistics, Intelligence, Ancient Knowledge, Mysteries, Sith Philosophy, Laws and Justice, Expansion and Diplomacy, Technology, and Biotic Sciences. The Councilors organize their responsibilities into various Ministries, which direct and employ the non-Sith officials.