1. Hafrada: Generally today, although not Halachically required, the custom is to separate the epithelium of the glans from the glans prior to the circumcision. This is done for medical and procedural reasons, including to ease the performance of the Peria and help with the healing.
2. Tefisa:[2] Prior to the circumcision, the Eiver is slightly hardened, and the foreskin is grasped and stretched above the head or glans [i.e. Atara] to prepare it for cutting away from the glans for reasons of safety to prevent injury. One includes with it some of the skin of the Peria. Enough of the foreskin must be pulled up and cut so the child is considered probably circumcised. This refers to the entire foreskin that covers the glans up until it’s ring.
3. Mogen: The foreskin that was grasped and stretched above the glans is then held in place using a specially made protective clamp [i.e. Mogen] which ensures that only this area of skin will be cut and prevents injury to the glans Chas Veshalom. There exist various different types of clamps, with some more tight than others, which prevent blood from coming out. Many of these more tight clamps have been prohibited by Gedolei Haposkim to be used due to various reasons.[3]
4. Blessings: The blessing over the circumcision is now recited by the Mohel. See chapter 9 Halacha 22!
5. Cutting of Mila:[4] After the blessing is recited, the foreskin that covers over the Atara and has been positioned above the Atara with the use of the clamp is now severed until the Atara becomes revealed. It is to be cut from the outer side of the clamp. There are different approaches regarding how the cutting is to take place, with some saying that it should be a form of pressing down with a knife [i.e. Derisa] and others saying it should be done with sliding the knife [i.e. Holacha]. Some say that one should do the cutting from below to above and not from above to below.
6. Blessing of father: At this stage, the father recites the blessing prior to the Peria. See chapter 9 Halacha 22!

Using nail: [5] The Peria is to be performed by the Mohel using his nail [and not by the use of scissors or any other utensil[6]]. [The Mohel uses two of his nail to cut the thin membrane covering the Atara glans until the skin of the glans becomes visible. This is done by entering the nails in between the membrane and glans. There are various methods of how this tearing of the membrane skin is to take place using the nails and as to the order of entering the nails.]

Tearing skin: The skin must be actually torn and it does not suffice to simply stretch it below the glans. After the skin is torn it is then stretched and folded over the shaft below the glans.

[1] See Pesakim Uteshuvos 264:10; See Encyclopedia Talmudit Vol. 46 Erech Mila p. 432-441; 462-469

[2] See Aruch Hashulchan 264:17

[3] See Igros Moshe Y.D. 2:119; Shulchan Menachem 5:130-132; Poskim in Pesakim Uteshuvos 264:10 footnote 11

[4] See Pesakim Uteshuvos 264:11

[5] Michaber ibid

[6] Bnei Tziyon 88 that performing it with one’s nails is a tradition from our forefathers which goes all the way back to Moshe, and is similar to the tradition of the identity of the four species used on Sukkos; See Pesakim Uteshuvos 264:13