As Lubos Motl stated by Luboš Motl in another answer, there is no consensus between Everett's interpretation contenders about what it means exactly. The common idea is that no state evolution other than unitary as per Schrödinger should be accepted (no collapse) but that's about it.
Indeed it is not clear at all what is supposed to be splitting or branching, and when. For a thorough review of this topic ("many what, exactly"?), see Multiplicity in Everett's interpretation of quantum mechanics (Louis Marchildon, 2015).