Quantum is the Latin word for amount and, in modern understanding, means the smallest possible discrete unit of any physical property, such as energy or matter. Quantum came into the latter usage in 1900, when the physicist Max Planck used it in a presentation to the German Physical Society. Planck had sought to discover the reason that radiation from a glowing body changes in color from red, to orange, and, finally, to blue as its temperature rises. He found that by making the assumption that radiation existed in discrete units in the same way that matter does, rather than just as a constant electromagnetic wave, as had been formerly assumed, and was therefore quantifiable, he could find the answer to his question.
Planck wrote a mathematical equation involving a figure to represent individual units of energy. He called the units quanta . Planck assumed there was a theory yet to emerge from the discovery of quanta, but in fact, their very existence defined a completely new and fundamental law of nature. Einstein's theory of relativity and quantum theory together explain the nature and behavior of all matter and energy on earth and form the basis for modern physics. However, conflicts remain between the two. For much of his life, Einstein sought what he called a unified field theory -- one would reconcile the theories' incompatibilities. Subsequently, superstring theory and M-theory have been proposed as candidates to fill that role.
Quantum is sometimes used loosely, in an adjectival form, to mean on such an infinitessimal level as to be infinite, as, for example, you might say "Waiting for pages to load is quantumly boring."