The ground-breaking developments of quantum mechanics in the early 20th century finally provided the answer to many outstanding questions about the nature and properties of the atom. Applying these principles to more complicated systems such as molecules and solid-state materials proved more difficult, however; even in classical physics there is no general solution to a three-body problem (such as the combined orbital motion of the sun, the moon, and the Earth), and to describe just a water molecule we have to deal with ten electrons and three atomic nuclei.
The success and applicability of DFT lies in some very clever realizations in the mid-1960s by Walter Kohn, Pierre Hohenberg, and Lu Jeu Sham. By not focusing on the individual electrons but instead using the electron density as the fundamental variable to solve for, and furthermore reformulating the many-body problem as an equivalent single-particle problem, density functional theory was born.
Over the following decades the method was turned into a practical tool by many contributors, and through the use of powerful numerical computers, DFT became an indispensable tool for materials science, chemistry, and many other fields. Relatively straightforward additions of the spin degree of freedom have enabled the description of magnetic systems, and on top of that, relativistic effects and even superconductivity can be treated.