Electrons, being fermions, can not occupy a state that a second one already has. Since they are spin-half particles, with +1/2 and -1/2, there can be two electrons in each state (with opposite spin).
The states correspond to solutions of the quantum potential well for different energy levels. (Ionisation corresponds to the energy required to escape a layer). The first layer is simply a spherical solution, 1s.
The second layer has a 2s solution, and N 2p (notionally a pair of lobes at right angles, so that there is one in the x, y, and z directions.
The 2sp layers can merge to give (N+1) lobes, being quite short at one end, and long at the other. This corresponds to lobes pointing to the vertices of a simplex. In carbon, these are occupied by a single electron each.
The layers of the third shell correspond to three solutions, 3s, 3p and 3d. d corresponds to something like 10 levels in N=3, but does not occur until after 4s (and before 4p. ie, The rows of the periodic table go
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1s (2 items)
2s (2) <---- gap ---> 2p [2N+2]
3s (2) <---- gap ----> 3p [2N+2]
4s (2) 3d (?) 4p
5s (2) 4d (?) 5p
6s (2) 4f 5d 6p
7s 5f 6d 7p
Atoms in 4f and 5f are usually shown below the main table (rare earth metals).