Bevel gears are used to change the direction of the rotation. They are used in watches, hand drills, vending machines, dentist's drills and electric typewriters. Usually the degree of intersection of the two shafts is 90 degrees, but it can vary depending on the angle that the teeth are cut. These gears can only be used with slowly rotating shafts.
Straight tooth bevel gears engage each other a whole tooth at a time--this is why they can only be used at low speeds. Bevel gears with spiral cut teeth engage each other gradually--first the edge and gradually the whole tooth. This makes spiral cut bevel gears better for high speeds--like bicycle and motorcycle gears.
The shafts of straight-cut and spiral-cut bevel gears are said to intersect because if the shafts were continued, they would intersect. The shafts of hypoid bevel gears do not intersect. They are used in the differential gears of automobiles so the drive shaft can intersect the differential gears below the level of the real wheel axis--leaving more room in the passenger compartment.