Here is another attempt to explain what happens at a bris, and what may happen at the bris.
During the bris, the mohel attempts to remove all of the ערלה (orlah)/foreskin, and as much of the עור הפריעה (or ha''priah)/ mucosal membrane (that lines the penis below the foreskin) as possible.
With a good חיתוך (incision) the proper amount of foreskin is removed and does not need to be revisited.
Sometimes "all" of the membrane is removed in that incision. Sometimes "some of it" is removed in that initial incision. Sometimes "none of it" is removed.
If any of the membrane remains, a good mohel will do "priah" - tearing and folding back the membrane beyond the glans before bandaging everything up.
Depending on how much of that membrane remains, as well as how it sits below the glans, it may need to be revisited (ideally immediately) either because the "bris" isn't kosher, or because it will be an unsightly eyesore.
So here is the bottom line:
Sometimes the bris goes well in one stage, as the foreskin and membrane are removed in one-fell swoop (so to speak).
Sometimes the foreskin is removed completely during the ceremony, but some of the membrane remains and needs to be removed in a second stage.
Obviously we would all prefer a one-stage and everything-goes-perfect-the-first-time bris milah experience.
But if a quick revisit, when the wound is still fresh, is necessary, especially if/not doing it could bring asthetic consequences, I make every effort to be up front, honest, and most importantly to take care of it right away so it never becomes an issue. As I've written elsewhere, I much prefer people be angry at me for five minutes than for 5 plus years.
Results matter, and I try to take all the necessary precautions so that the results are pleasing and pleasant for a very long time.