Quantum mechanics can be summarized by the following:
Each particle or combination of particles (including the universe as a whole!) has a complex-valued function, usually denoted Ψ (Psi), which contains all of its state information. This function is also called the "Wavefunction" because the constraint on its time-evolution is a Wave Equation - Schroedinger's Equation (or, for taking into account special relativity, various field equations such as the Dirac Equation)
These functions can be added to each other. That is, it is possible to add one state to another state; the result is a different state which is in a very specific sense part way between the two you combined. Moreover, if you take one state and let time progress in it, and you take a second state and let time progress in it, then you add the states together, you get the same result as adding them together in the beginning and then let time progress.
Importantly, you can also split a state up into different parts. Slicing the states in different ways yields all of the interesting part of the theory. In particular, one often chooses a way of slicing the states up so that some quantity - position, momentum, energy, angular momentum, etc. - has a specific, different value for each slice.
The measure of a state is its squared amplitude. So, the expected value of some physical quantity can be found by slicing the state up so each slice has a specific value of that quantity, then getting the measure of each slice, and multiplying it by the value for that slice, and adding it all up. Alternately, you can stop short of adding it all up to get the distribution over those values for the full state.
One thing that's important is, you can't keep slicing to narrow things down - at least, not without losing the meaning of the old slices. A slice that has perfectly defined position, say, does not as a matter of mathematical identity have a perfectly defined momentum, even if you sliced it to perfectly define momentum first. Indeed, such a slice is a pure mix of all possible momenta. If you try to slice this one point function slice down to get specific momentum values, then the resulting slices are no longer confined to single points in space. This is what is known as the Heisenberg Uncertainty Principle.
There are other confusing results. The most illuminating is the two-slit experiment. But even short of that, you run into troubles of interpretation. If you shoot an electron at a barrier, part of the wave is reflected and part of it continues onward. What do you say happened? The electron was split in two parts? Sure, of course. You have a slice of it that was blocked and a slice that was permitted. But, well, no -- if you actually look, then you will find it entirely reflected or entirely permitted. However, if you repeat the process many times, the number of outcomes will be proportional to the two corresponding slices of the wave. This is known as the Born Probability Rule. What can we make of that?
It seems that we only get to perceive one slice. Yet, what sort of slice it is varies from case to case in a way that is on the surface baffling (it takes a very complicated theory to make sense of this).
Interpreting Quantum Mechanics is something even a lot of people who use QM in their research simply don't have the time to deal with, since it can become somewhat of a philosophical obsession. The Copenhagen interpretation is very unsatisfying, but most physicists have aesthetic and/or comprehensional problems with the main alternative, the Many-Worlds Interpretation. While nonlocal-variable deterministic theories work and are not particularly mind bending at first glance, they are inelegant and the least popular of all.
Local Hidden Variable deterministic theories were ruled out by experiments in the 80's which took advantage of the Bell inequality, which showed that they were not equivalent to the normal formulation of QM and pointed out a difference. The experiment was done, and Local Hidden Variables were completely ruled out.
One thing that can clear up a little of the mess is this second example: Instead of shooting an electron at a barrier, we shoot a cannon ball at a barrier. Using a quantum calculation, we would either determine that the vast bulk of the wavefunction of the cannonball would bounce off the barrier, or conversely determine that the vast bulk of the wavefunction of the cannonball would penetrate the barrier. While in a technical sense the wavefunction is still split between permission and reflection, due to the tremendous mass of the object in question, the non-favored outcome is extremely strongly suppressed -- not a factor of a ten or a thousand in magnitude, but a factor of a ten to the thousandth power (to give a conservatively SMALL estimate). So it seems we could just say, "Oh, phew! All the quantum effects go away on their own for heavy objects!" But the problem is, they don't ALL go away. See Schroedinger's Cat.
One common misperception of quantum mechanics is that it always limits energies to particular discrete levels. It does do this whenever particles are trapped in a potential well. When particles are free, their energies may vary continuously. However, even in this case energy exchange is mediated in quanta, which is to say in discrete chunks. The size of these chunks can be any value, but still, the rules prohibit truly continuous energy transfer. The number and size of chunks it's sent in can be fuzzy in such a way that is effectively continuous (dropping an apple, for instance, would involve a transfer of this type)
Though Quantum Mechanics, via the Schroedinger equation, reproduces Newtonian mechanics for reasonably heavy objects, and in the formulation of Quantum Field Theory has incorporated the predictions of Special Relativity, QM has so far resisted all attempts to reconcile it with General Relativity. Though it is possible to define wave functions in curved space-times, these efforts have not yet succeeded completely. A complete solution would have several notable difficulties - for example, making the definition of space-time itself a function of the wavefunctions. This is especially tricky since superposition of wavefunctions would cause different mass distributions, which would in turn cause different definitions of space-time, so you end up in the sticky position of adding functions which are not defined on the same domain. So long as the mass distributions only stretch space, it is possible to resolve this with greater abstraction in the mathematics -- but if mass distributions cause space's topology to change (e.g. wormholes), that is extremely awkward. Also, since gravitation is so weak compared to the other forces (only being significant for large masses), it has so far resisted attempts to measure it on the scale of individual particles. So, even if we could rule out topology changes, it would be very difficult to verify any theory connecting the two.
In the discussion above, I may have inadvertently given the impression that QM sprung fully-formed from the forehead of Schroedinger. This is utterly false, and the history writeup of bigmouth_strikes, below, lists a few of the milestones.