The egg tooth is a small, pointed projection on the top of the beak, near the very tip. It is made of the same hard material as the beak.
The egg tooth appears on the baby bird's beak about Day 7 in its development inside the egg. When the bird is ready to hatch, the egg tooth is like a little can opener. The chicks will pip the shell (make a little hole), and chip away at the shell as it moves around the shell.
After the chick hatches, the egg tooth only lasts a couple of weeks before it falls off or is reabsorbed.
I think that they'll grow back a little bit but not fully. It depends how much is taken off. Debeaking is usually done to commercial laying hens to keep them from cannibalism because they are in such small cages with each other.