The Lt. Governor replaces the Govenor
Lt. Gov.
The Lt. governor.
If the governor of Georgia dies, the lieutenant governor automatically assumes the role of governor. This succession is outlined in the state's constitution. If both the governor and lieutenant governor are unable to serve, the line of succession continues to the speaker of the Georgia House of Representatives.
the Vice president
it dies
The Secretary of State assumes the power of the Governor when the current Governor dies, resigns, or leaves office for other reasons. The current Secretary of State in Arizona is Jan Brewer, a Republican. The current Governor of Arizona is Janet Napolitano, a Democrat, who is currently being considered to become part of President-Elect Obama's cabinet.
If the governor of Texas dies, resigns, or is removed, the lieutenant governor will take his place until reelection.
extinction
The Governor can appoint someone to take his place until the next Federal election comes about.
According to the Idaho Lieutenant Governor's website (refer to the link, below), the Lt. Gov. would take over as Governor in the event that the Governor leaves the State or is incapacitated or worse.
the vice president takes his place