Quantum Mechanics is a theory which says that when you make a measurement of a particle, there are only certain results you can get for an answer. An analogy would be the number of cents you have in your pocket. You can only have money in increments of one penny - you can never have 23.14789 cents. Instead, the number of cents always has to be a round (whole) number.
This sounds natural for money - after all, there's nothing smaller than a penny - but it was a very surprising discovery in the early 1900's when it was found that physical measurements behave this way, too. For example, the electron in a hydrogen atom was found to exist only in certain energy levels - or "quantum states". When an electron moves from one energy level to another, it doesn't gradually pass through all the energy levels in between. Instead, there is a "quantum leap", and the electron instantly leaps from one energy level to the next. In general, a quantum leap is when a particle changes from one quantum state to another.
This particular example of the electron in a hydrogen atom can be measured by looking at the light given off when the electron "leaps" from one quantum state to another. If an electron drops to a lower energy state, conservation of energy demands that the extra energy has to go somewhere. In this case, the extra energy goes into a "photon", or a light particle. The frequency of the light tells how much energy is in each photon, which in turn tells us about what the electron is doing. Here is an image which shows the different frequencies of light that are emitted by Hydrogen atoms:
As you can see, the light only comes off in certain frequencies, which means that the electrons are only in certain energy states. This was one of the pieces of evidence found in the early 1900's which convinced physicists that nature really did have a "quantum" aspect.
The above image is taken from a web page which has much more information about the spacing of the lines and the quantum theory which explains it.
But is it still possible that the electron is actually moving through all the energy levels in between, but doing it too fast to see? Probably not, or the photons which are detected would have a different character - they would not be a single particle but rather a bunch of different particles slightly spread out in time. Because the photons behave as single particles (you can't split them up into smaller sub-particles), it seems that the "quantum leap" really is happening. (source unknown)
15.13 - Dissociating Water Acoustically - Liberation of Quantum Constituents
3.22 - Quantum Leap Delta equivalent to Locked Potentials Delta
4.10 - Component Dynamics of Quantum Construction
7B.08 - The Etheric Quantum Soup
Figure 3.37 - Successive Centralizations or Quantum Leap
Table of Quantum Particles