This is very simple, so I'm not going to spend a lot of words on it. Reins are [typically] leather straps that attach to a bit and bridle, for the purpose of steering a horse. A reign is a period of rule by a monarch. You can reign in a country, but you cannot reign in a horse. There's not enough room in there. When someone says they want to "reign in" some kind of activity by someone else, or "seizing the reigns of power" they are demonstrating their poor command of the English language and you should stop reading immediately before you get any on you. Don't be that person.