Early in the twentieth century it became evident that the state of very small systems did not change in a continuous way. Atoms jump from one allowed state to another, skipping over the forbidden states in between. Newton's laws of classical mechanics are not equipped to handle situations like this so a new approach was required. Quantum Mechanics evolved to extend classical mechanics to very small systems.

Quantum mechanics is intensely mathematical. Our everyday experience provides no intuition in the quantum realm. No one knows what reality looks like on the quantum level, or if in fact there is any quantum reality. What we do know is that applying the techniques of quantum mechanics to molecules, atoms, sub-atomic particles and fields has produced predictions that have been thoroughly verified by experiment. Quantum electrodynamics, the interaction of electromagnetic waves and matter, has proven correct to one part in a million million.

In addition to the quantization of certain system properties, quantum mechanics addresses other phenomena where classical mechanics fails. These include wave-particle duality, the uncertainty principle and quantum entanglement. As in the case of relativity, I am going to make no attempt to derive results from theory. For our purposes we can skip a lot of details and look at the big picture. The numbered statements below summarize some aspects of quantum mechanics.

1. It turns out that all objects of any size follow quantum mechanical rules. We got away for so long with our ignorance of quantum mechanics because quantum mechanical states fall closer and closer to each other as system size increases. In the limit of visible systems the states appear to form a continuum and the rules of classical mechanics may produce useful predictions.

2. There is a mathematical entity called the quantum state vector (or wave function) that contains the information that makes up the state of a system. The way the state vector varies with time is expressed mathematically in a function that combines all possible states of the system.

3. From the time evolution of the quantum state vector, a predicted state may in principle be calculated for any future time . This predicted state is only an expectation value, the average of values actually observed in duplicate observations of identical systems. The act of observing an atom sized system disturbs the system, knocking it into a state other than the expectation value. As system size increases the difference between the actual system state observed and the expectation value decreases. The process of extracting an actual state from the function combining all possible states is called quantum state reduction (or wave function collapse). There is an element of chance in the actual state in which a system is found following quantum state reduction.

4. Historically we thought of waves as one thing and particles as another. Quantum theory mixed these things, somewhat analogous to the way relativity mixed space and time. Light has both wave-like and particle-like aspects. Electrons have both particle-like and wave-like aspects. Which aspect is presented to us depends on what experiments we perform on the objects.

5. Certain pairs of quantum system variables, called conjugate pairs, are related through the Heisenberg uncertainty principle. The precision with which these pairs may be known is limited by the relationship: (uncertainty of variable 1)X(uncertainty of variable 2) = 5.5×10^-33 joule seconds. Conjugate pairs include, position-momentum and energy-time among others.

6. From Wikipedia: "Quantum entanglement is a property of the state of a quantum mechanical system containing two or more degrees of freedom, whereby the degrees of freedom that make up the system are linked in such a way that the quantum state of any of them cannot be adequately described independently of the others, even if the individual degrees of freedom belong to different objects and are spatially separated."

See the Wikipedia article for more - much more - on quantum mechanics.