electron shells
Electron shells are the grouped energy levels that electrons occupy around a nucleus, each shell labelled by the principal quantum number n. Historically they were also named with letters — K for n equal to one, then L, M, N — and they correspond roughly to layers at increasing average distance from the nucleus. Each shell sits at a higher energy than the one inside it and can hold a fixed maximum number of electrons.
The capacity of a shell is not arbitrary; it falls straight out of the quantum numbers. Counting all the allowed orbital shapes and orientations within shell n, and doubling for the two spin states, gives a capacity of two times n squared: two electrons in the first shell, eight in the second, eighteen in the third, and so on. This counting, together with the exclusion principle, is why atoms fill up the way they do.
Shells are a wonderfully useful picture, but they are an organising scheme rather than literal physical layers. Within a shell the electrons spread into orbitals of different shapes that overlap and interpenetrate, and in heavier atoms the energies of subshells from different shells can even cross over. Still, the shell idea captures the essential rhythm of atomic structure and underlies the whole logic of the periodic table.
Each shell n holds at most two times n squared electrons, once spin is counted.
A shell's capacity of 2n² is the maximum, not the order of filling. Because subshell energies overlap in heavier atoms, outer shells often start filling before an inner one is full.