How many electrons in each layer of an atom?

Answer 1

The Nth "shell" of an atom can hold 2N2 electrons. So 2 in the first, 8 in the second, 18 in the third, and so on.

(Both "layer" and "shell" are a bit misleading, though "shell" at least is probably too historically ingrained to eliminate.)

Answer 2

The first energy level can hold up to 2 electrons, the second energy level can hold up to 8 electrons, the third energy level can hold up to 18 electrons, and the fourth energy level can hold up to 32 electrons.