A duchy is an Amtgard chapter larger than a barony, but smaller than a Principality or Kingdom. A barony must hold an average attendance of 30-40 people during a six month period in order to retain its status. The exact size required to become a Duchy varies from kingdom to kingdom.
What can a Duchy do?
A Duchy is allowed to give ladder awards up to the 6-8th level, depending on their kingdom's corpora. The ability to grant noble titles and class masteries varies between corporas, with most reserving this right for kingdoms.
The elected leader of a Duchy is a Duke or Duchess. 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 exclusively elected, though some corporas require that a champion 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.
The next step
Duchies that meet their kingdom's requirements (usually 60+ people a week for a period of time, as well as other requirements) can be elevated to a principality.