Horses and kings, while having some similarities, are not the same things. Perhaps that is a common misconception that leads to the confusion over the words reign and rein. But I doubt it.
Reign is the period of time a king (or other sovereign leader) occupies the throne (or other way of expressing the seat of power in a government). It can also be used as a verb: to rule, or possess sovereign power or authority.
Rein is that thing that holds horses back.
In The Return of the King, Aragorn does a lot of reining of horses, but takes his sweet time getting around to doing any actual reigning over the kingdom of men.
(Weak example, I know; I just wanted to look at Viggo Mortensen. You're welcome.)