Both of these words can each appear as either noun or verb.
The noun rein is almost always plural. Reins are a pair of straps used to control horses pulling a horse vehicle, such as a carriage or wagon. One end of each rein connects to the metal bit in the horse’s mouth, and the other end of the rein is held by the driver, who guides the horse by pulling on the reins.
The verb to rein, is used mainly in the expression to rein in—which means to bring something under control:
The driver used the reins to rein in the horses, who were out of control.
Some teachers rein in students by threatening to take away recess.
The verb to reign means to rule, as a king or queen rules—to hold power:
The queen reigned gracefully for seventy years before she died.
The noun reign refers to the period of a king or queen’s ruling:
The emperor’s reign lasted for fifty years.
Under the queen’s reign, people were happy.