A rein is “the part of the bridle, which extends from the horse’s head to the driver’s or rider’s hand,” as Dr. Johnson defines it. You use it to control the horse.
A reign is the “time of a king’s government,” or the rule of any person in authority: the reign of Queen Elizabeth II, the reign of Pope Benedict LXXXIV, the reign of J. Rutherford Pinckney as president of the Schenectady Small Arms & Biscuit Co.
These two words are often mixed up, especially when we’re speaking metaphorically instead of talking about actual horses or kings.
But it’s not hard to sort them out. Almost all the metaphors and idioms that use the word rein refer to horses, so they need rein without the G.
To give someone free rein is to let him do what he likes, without control, as you might slack off on a horse’s reins when you were confident that he knew the way. “Jennings comes to us highly recommended, so we’re giving him free rein in Marketing.”
To take the reins is to take control. “A lot of things have changed in Marketing since Jennings took the reins.”
To rein someone in is to take control of him, perhaps after having been appalled by the results of giving him free rein. “Jennings killed three interns in Marketing, so we had to rein him in.”
We use the word rein without the G even if we’re talking about controlling kings and queens. “The reign of Charles I was an era of constant conflict. The king insisted on giving his whims free rein, no matter how often Parliament tried to rein him in.”