Promotion Branching is a mechanic that appears in Fire Emblem Gaiden, Fire Emblem: The Sacred Stones, Fire Emblem: Shadow Dragon, Fire Emblem: New Mystery of the Emblem, Fire Emblem Awakening and Fire Emblem Fates. It allows units to have two or more options of classes to promote to. For example, in The Sacred Stones, a Cavalier can promote into a Paladin or a Great Knight.
In Shadow Dragon and New Mystery of the Emblem, Pegasus Knights usually promote into Wyvern Lords, but with an Elysian Whip purchased from the Online Shop, they can promote into a Falcon Knight instead.
In Awakening and Fates, promotion branching functions in the same way as it does in The Sacred Stones, with different classes. Likewise, in Fates, the only Lord class that allows branched promotion is the Nohr Prince(ss), though this aspect is only included in the Revelation route.
For details on what classes' options are, see the respective class listings: