Like a lot of commonly confused words, reign and rein actually have similar meanings as well as spellings. Reign refers to having authority or rule; rein alludes to the straps used to guide horses, therefore also implying control.
It’s easy, then, to mix up expressions that use these words. “Free reign” makes sense to me, but the correct use of the phrase according to Webster’s (and AP) is “free rein.” Likewise, if someone is exerting control, or holding back, they’re “reining it in,” and someone seizing control is “taking the reins.”
“Reign” tends to be used less often, but you need it when referencing the time someone was in power or highly influential (i.e. the reigning champion). It can also denote a prevailing emotion: “Fear reigned in the aftermath of 9/11.”