Overview
Praefectus, often with a further qualification, was the formal title of many, fairly low to high-ranking, military or civil officials in the Roman Empire, whose authority was not embodied in their person (as it was with elected Magistrates) but conferred by delegation from a higher authority. They did have some authority in their prefecture such as controlling prisons and in civil administration.
Praetorian prefects
The Praetorian prefect (Praefectus praetorio) began as the military commander of a general's guard company in the field, then grew in importance as the Praetorian Guard became a potential kingmaker during the Empire. From the Emperor Diocletian's tetrarchy (c. 300) they became the administrators of the four Praetorian prefectures, the government level above the (newly created) dioceses and (multiplied) provinces.
Police and civil prefects
Praefectus urbi, or praefectus urbanus: city prefect, in charge of the administration of Rome.
Praefectus vigilum: commander of the Vigiles (firemen and police).
Praefectus aerarii: nobles appointed guardians of the state treasury.
Praefectus aerarii militaris: prefect of the military treasury.
Praefectus annonae: official charged with the supervision of the grain supply to the city of Rome.
Military prefects
Praefectus alae: commander of a cavalry unit.
Praefectus castrorum: camp commandant.
Praefectus cohortis: commander of a cohort (constituent unit of a legion, or analogous unit).
Praefectus classis: fleet commander.
Praefectus equitatus: cavalry commander.
Praefectus equitum: cavalry commander.
Praefectus fabrum: officer in charge of fabri, i.e. well-trained engineers and artisans.
Praefectus legionis: equestrian legionary commander.
Praefectus legionis agens vice legati: equestrian acting legionary commander.
Praefectus orae maritimae: official in charge with the control and defense of an important sector of sea coast.
Praefectus socium (sociorum): Roman officer appointed to a command function in an ala sociorum (unit recruited among the socii, Italic peoples of a privileged status within the empire).
From Wikipedia (CC BY-SA 4.0).