The office was used to describe an official appointed by the English monarch to oversee English colonies. Because former English colonies have undergone different constitutional history, the name now refers to officials with radically different amounts of power.
In Britain's remaining crown colonies the governor may be a direct appointee of the British government and play an active role in lawmaking (though usually with the advice of elected local representatives).
In the United States, "Governor" refers to the chief executive of each state. In all states, the individual is directly elected and has considerable practical powers, though this is moderated by the state legislature and in some cases by other elected executive officials. In colonial America, the governor was the representative of the monarch who exercised executive power. During the American Revolutionary War, the royal governors were all expelled, but the name was retained to denote the new elected official.
In Australia, each state has a governer as its formal head. In theory state governers are appointed by the queen and act as her representative. In practice they are always appointed on the advice of the state premier and play a purely ceremonial role. State governers have emergency reserve powers but these are almost never used.
Related offices include the Governor-General which refers to the de facto head of state of Australia, Canada, Jamaica, and other Commonwealth Realms who nominally represents the British monarch. In theory, they have almost dictatorial powers, but in practice they are considerably circumscribed by convention. Governors General are appointed on the advice (which is inevitably followed) of the prime minister of the nation.
Another related office is the lieutenant governor which in a state of United States is the second highest executive official in a state and which in Canada is an office for a Canadian province which is analogous to the governor for an Australian state. In Australia the state lieutenant governors have no powers but stand ready to take up the governer's role in case of illness or inability; the Commonwealth Lieutenant Governor is, by convention, the most senior State Governer. Australian states also have a third office, which is subordinate to the Lieutenant-Governor, the "Administrator" (who acts as Governor if both the Governor and Lieutenant-Governor are unable to perform their duties). The offices of Lieutenant-Governor and Administrator are both of very little practical relevance.
The difference in terminology between the Australian case and the Canadian case is significant. In the Australian case, the governor nominally derives power directly from the monarch and is in practice nominated by the Premier of a state. In the Canadian case, the lieutenant governor nominally is appointed by the Governor-General and in practice is named by the Federal Prime Minister.