What is an intuitive explanation for Electron Configurations
Electron configurations are a bunch of numbers and letters, and they also follow a pattern. But there is also a logic behind the whole thing.

Basically, chemists use electron configurations to see in which atomic orbitals the electrons are found. From this several things can be derived, most of all the reactivity of said electrons (which is where the chemistry happens).

The intuitive answer is this: Electron configurations are a way of telling in which atomic orbitals the electrons are put in, according to the Aufbau principle and the Pauli exclusion principle. Decoding it is pretty easy, as it always follows this pattern:
$n$$(s|p|d|f)^x$ where $n$ is the main quantum number (or shell number), $(s|p|d|f)$ denotes the atomic orbitals with the quantum number and $x$ is the number of electrons found in the orbitals.

The orbital "groups" ($p$, $d$, $f$) contain several "orbitals" (side note: an orbital is strictly seen a one-electron function) where two electrons with opposite spin fit in. $p$ has 3 ($p_x,\; p_y,\; p_z$), $d$ has 5 ($d_{xy},\; d_{xz},\; d_{yz},\; d_{x^2-y^2},\; d_{z^2}$) and $f$ even more. This means that the maximal number of electrons in the $p$ group is $6$, and in the $d$ group $10$. So when you fill in the electrons according to the aufbau principle, you have to go to the next orbital group when you reach this number.

Bottom line: Every electron in an atom is defined by the four quantum numbers $n$ (main or principal quantum number), $l$ (azimuthal), $m_l$ (magnetic) and $m_s$ (spin). Using the electron configuration we can derive the first three of these four numbers.