The current definition of the Hebrew calendar is generally said to have been set down by the Sanhedrin president Hillel II in approximately AD 359 (4119). The original details of his calendar are, however, uncertain.
The Hebrew calendar is used for religious purposes by Jews all over the world, and it is one of the official calendars of Israel (the other one being the Gregorian calendar).
The Hebrew calendar is a combined solar/lunar calendar, in that it strives to have its years coincide with the tropical year and its months coincide with the synodic months. This is a complicated goal, and the rules for the Hebrew calendar are correspondingly fascinating.