There are two key ideas associated with DFT.
1. A theorem.
The ground state energy of an interacting electron gas is the minimum value of a unique functional of the charge density n(r) in the system.
This is an exact result.
The problem is that to determine the exact density and energy one needs to know the "exchange-correlation" functional.
2. An approximation.
One can make a local density approximation (LDA) to the exchange-correlation functional so that the density is written in terms of a set of "orbitals" that are found by solving a set of self-consistent equations that have a mathematical structure similar to the Hartree-Fock equations for the same system.
These distinct ideas are respectively associated with two different papers, published 50 years ago. The first is by Hohenberg and Kohn. The second is by Kohn and Sham.
Aside: The history and significance of these papers has been nicely summarised recently in a Physics Today article by Andrew Zangwill.
I think the community needs to be more precise when they talk about DFT.
Broadly speaking, some people in the chemistry community give me the impression that they think if they can just tweak the parameters in their favourite exchange correlational functional then they are going to be able to get agreement with experiment for everything.
In contrast, consider this paragraph from the introduction of a recent physics paper:
Density functional theory (DFT), in essence a sophisticated mean field treatment of electron-electron interactions, provides a very good approximation to the interacting electron problem, enabling the theoretical description from first principles of many properties of many compounds. However, DFT does not describe all electronic properties of all materials, and the cases where it fails can be taken to define the “strong correlation problem.”Surely, it would be better to replace DFT here with DFA=Density Functional Approximations.
Aside: I should say that besides this paragraph I really like the paper and the authors.
The distinction I am making here was particularly stressed in a recent talk I heard by Tim Gould.