The name Gregory is pronounced GREG-ory. It is originally a Greek name, Gregorios, which means “watchful” or “vigilant” (the modern Greek version is Grigoris, which is also associated with the word for “swift.”) This developed into the Latin Gregorius, which was very popular among the early Christians, partly because St Paul had urged believers to be vigilant in the New Testament. The name kept its religious connotations for centuries; it was given to several saints in the first few centuries after Christ, and there was a famous Pope Gregory in the 6th century.
It is a particularly widespread name in Scotland, where it is sometimes given as Gregor (which is also the German version; the Scandinavian is Greger or Gregers.) The usual short form is Greg.