A Barony is typically allowed to give ladder awards up to the 4th or 5th level, depending on the kingdom it's apart of. A barony is not typically allowed to bestow titles of nobility.
The elected leader of a Barony is a Baron, or the feminine Baroness. Other offices that may or may not be filled due to the needs of the park are Regent, Prime Minister (or Chancellor), and Champion. Most of these offices are elected, sometimes with the exception of the Champion, who can be chosen by right of arms.
In most places candidates declare and then a quals is held. Qualifying candidates are then voted on by either the dues paid populous or all in attendance, depending on the rules of the park. Some kingdoms do not require baronies to hold a qual for office.
The next step
Baronies that meet their kingdom's requirements (usually 30-40 people a week for six months) can be elevated to a duchy.