What is $\psi$ and what is $|\psi|^2$? What do they represent?

(N/A) $\psi$ is the wave function. It is a mathematical function and does not have any direct physical significance.
$|\psi|^2$ represents the probability density of finding an electron at any point in the atom. It has physical significance. The probability of finding an electron is proportional to the value of $|\psi|^2$.