Top: 3-meter-tall sculpture of Einstein's formula at the 2006 Walk of Ideas, Germany.
In relativistic thermodynamics, E = mc² is an equation which defines the principle of mass-energy equivalence, which states that energy is proportional or "equivalent", according to the speed of light squared, to matter: [1]

$E = mc^2 \,\!$

Said another way, mass and energy are different manifestations of the same thing.

The now-famous expression was developed proposed 1905 by German-born American physicist Albert Einstein in his paper "Does the Inertia of a Body Depend upon its Energy-content?", wherein he argued that the equivalence of mass and energy is a general principle, which is a consequence of the symmetries of space and time: [2] In his own words, as voiced in the adjacent video:

“It followed from the special theory of relativity that mass and energy are both different manifestations of the same thing. A somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m , in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned. This was demonstrated by Cockcroft and Walton in 1932, experimentally.”

German physicist Ingo Muller argues that E = mc², defining energy, and S = k ln W, defining entropy, are the two most famous expressions in all of science. [2]

References
1. Bodanis, David. (2000). E = mc² - a Biography of the World's Most Famous Equation. New York: Berkley Books.
2. Muller, Ingo. (2007). A History of Thermodynamics - the Doctrine of Energy and Entropy, (ch. 10: Relativistic Thermodynamics, pgs. 289-305). New York: Springer.
3. Einstein, Albert. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" (See also the English translation), Annalen der Physik 18: 639–643