A governor was the head of government for a colony, province, or other sub-national body politic in the galaxy. There were several usages of the word throughout the Galactic Republic, Galactic Empire, New Republic and Galactic Alliance.
Old Republic governors
Governors existed in the Galactic Republic, though it was a federal system, therefore control of local systems and sectors were left to their own governments. It seems that the Old Republic only used governors temporarily, for systems that were unstable or for those that were newly-admitted to the Republic and lacked any capable governing system of their own. Mousul sought to get appointed sector governor of Ansion's region of space.
The one known example of an Old Republic governor was in the Seswenna sector, which was ruled by a governor and a lieutenant governor from at least 33 BBY until the end of the Republic. The governor and lieutenant governor of the sector also simultaneously served as the governor/lieutenant governor of its capital planet, Eriadu.
The Empire left the self-government of many planets intact, especially those that were either indifferent or supportive of the New Order (such as Naboo and Corellia). Less than 1/80th of all planets had their governments altered. Planets that rebelled, were deemed hostile to the New Order, or newly-annexed by the Empire, had governors appointed. Certain worlds that had senators who were deemed less than loyal to the Emperor were sometimes assigned governors, despite the opposition of their senator. This was the case on Chandrila, when Jesco Comark was appointed as Imperial Governor to keep watch on Senator Mon Mothma. However, pressure by the populace, who were loyal to the more experienced Mothma, caused Comark to abandon the post.
Other worlds, which were further away from Coruscant, such as Bakura, retained their government with only minor changes, but also had an appointed governor. The Empire also kept the position of Lieutenant Governor. Except for a few governors chosen by Palpatine, Imperial Governors were selected by the Imperial Ruling Council and received most orders from the Diplomatic Services. Emperor Palpatine had vague plans for eventually replacing the planetary governors with Dark Side Adepts, creating a "Dark Empire".
New Republic governors
The New Republic's policy on governors (termed Governors-General) seemed to mirror that of the Old Republic. Both Corellia and Rhinnal had Governors-General, and at least Corellia's was elected by the planet's population. The Galactic Alliance continued the system, although several years after the end of the Yuuzhan Vong War it removed the Governor-General system from Corellia.