Dimension Lords are almost always powerful sorcerors, deities (of at least greater rank and beyond), eldertich abominations, as well as a few exceptional mortals who have ascended beyond their former state.
The criteria is simply being the most powerful being in their entire universe. Each is a being of power so vast that even gods bow (or tremble) before them. The mightiest exercise complete control over their respective universe and can manipulate everything within them including e.g. time, space, life, matter, energy, etc.
They may be highly advanced forms of life, or they can be cosmic deities as old as time.
Some are benign beings, but sadly the usual cases are Dimensional lords that are nearly always motivated by conquest, seeking to add more real estate to their territory. All have at least some sort of responsibility to control, monitor, and/or observe their respective realities.