To explain the spectrum of radiation from bodies, Max Planck in 1900 introduced the concept that light occurs in discrete bundles of energy, or "quanta." In 1905, Albert Einstein proposed that these quanta were actual particles, now called photons. He explained the photoelectric effect in which light produces an electric current. In 1913, Niels Bohr used quantum theory to derive the energy levels of the hydrogen atom, arguing that the electron circles the nucleus in only certain, discrete orbits.
In 1923, Louis de Broglie proposed that electrons had wave-like properties, which was soon confirmed in experiments. This lead to the "wave-particle" duality in which objects are somehow regarded as being both waves and particles simultaneously. This lead further, in 1925, to the development of more formal mathematical theories of quantum mechanics, first one by Werner Heisenberg and then another by Erwin Schrödinger. Schrödinger introduced the idea of the wave function and the famous equation that bears his name.
Heisenberg proposed the equally famous Heisenberg uncertainty principle, which says that certain properties such as momentum and position cannot be measured simultaneously with unlimited precision. This implied the breakdown at small distances of the classical, Newtonian picture in which the motion of a body is completely determined by its initial conditions and the forces acting on it.
Quantum mechanics introduced a randomness into the universe that is still being argued over. It also forced us to rethink some of our common conceptions of the nature of reality. However, two notions are frequently misunderstood and misused.
First, quantum mechanics did not show that Newtonian mechanics was "wrong." Most physical phenomena, including most of what goes on inside living bodies, can be understood with Newtonian mechanics. Quantum mechanics only comes in at very small distances, such as inside atoms, or in very special macroscopic circumstances, such as superconductivity.
Second, those quantum phenomena that seem to violate normal common sense can still be understood in purely material, physical terms, and with a slightly more developed common sense. Although mystical elements have been proposed, nothing we know about the quantum requires us to introduce such elements. And the bottom line remains: quantum mechanics agrees with all observations.