The first two stages of a "Kosher" Bris include the removal of the foreskin and the removal of a mucosal membrane which is present directly below the foreskin.
Milah is the act of circumcision, which specifically targets the skin of the foreskin.
Priah (uncovering) is the act of the removal of the membrane.
While Milah is virtually done the same way by most practitioners, i.e. with a knife (though see below), Priah is done differently by varied practitioners, because some aim for what's called "Milah U'Priah B'vas Achas" (essentially doing both actions in one action), while others aim to make them separate actions. (see below as well)
It is important to note that for all practitioners, the amount of foreskin removed ends up being determined by a number of factors. They include: the baby's unique anatomy (some have a larger shaft, some a smaller shaft, some a larger glans, some a smaller glans, some are born with partial foreskin or even with no foreskin!), the mohel's estimate, the heat of the moment.
Due to the first issue (unique anatomy), and concerns over the third issue (heat of the moment), I use a marker in my estimation (second issue) to properly give the time to make sure the assessment of the estimate is accurate, and also to alleviate the possibility of estimating incorrectly in "the heat of the moment." A general principle to follow is (counterintuitively), the longer the shaft the less foreskin, the shorter the shaft, the more foreskin. [Though it is important to remember that ultimately every baby is different and hopefully the mohel is guided by Hashem and with common sense and training to get the right amount.]
With respect to the membrane, if it is removed with the removal of the foreskin, that element of the bris is complete as well. If it is not removed at that time, then what follows is the mohel following his tradition, in one of the following options:
1. Tearing and folding back the membrane
2. Removing the membrane with scissors or the "milah knife."
Sometimes the removal of the membrane is necessary for halakhic reasons, sometimes for aesthetic reasons.
No matter how the mission is accomplished, the hope is that all should look nice and should heal beautifully. May it be so every time!