Here's a story about Quantum mechanics:
Once upon a time there was a man named Schrodinger. He was a pretty smart cookie, and he invented what is known today as the Schrodinger Equation.
He wanted to prove this equation, so he set out to make an experiment: The cat in a box experiment.
Step one, put the cat in the box, close the box.
Step two, make a small hole in the box.
Step three, insert a gun through the hole, pull trigger.
Step four IMPORTANT!! DO NOT OPEN THE BOX!
Apart from the screeches of terror from the cat, Schrodinger had no way of determining whether the cat was dead or alive. So yeah this is how he proved how quantum mechanics works on probability of particle placement in a system instead of where it actually is, because determining it's exact position cannot be done. Rather, you can determine areas where it probably is.
General craziness aside, Quantum mechanics is scary to many people simply because it is so loving mathematically involved. You need to be a master of Calculus to even dream to understand the mathematical relationships used in Quantum. Wrapping your head around the general behavior of particles isn't too bad, but when you apply math to that behavior...
But I digress. The most basic math is quite simple. In quantum, your first priority is the general Location of things. Location, location, location, where the forget is that particle at right now? Is it over there? Did it somehow get passed that barrier of 0% probability and is now over here? You can't know for sure, but you can get accurate approximations using Math. All quantum systems can be represented by a function denoting the particle in that system. The function comes in two flavors, Time dependent and Time independent. Fun fact: Multiplying the Time dependent function with it's Time independent counterpart should yield 1. This means the probability of finding the particle in the system is 100%. If the value is 0, then there is 0% chance the particle is in this system at any given time. Generally for it to work, you kind of want your interesting little particle to be present in the system at all times, so it should be 1.
Then you have the juicy Add-ons you can put on these functions, the so-called Operators. These are the observable characteristics of the system, that is, specific attributes the system has (like energy, momenta, etc.). You can think of these guys as Black boxes. You put your original function in the black box, the black box will spit out a new modified function. So from the general function denoting possible location, you can get such things as the Energy characteristics of the particle at certain positions, it's velocity at certain positions, it's moment of inertia at certain positions, etc. You guys are starting to get the idea.
The term Quantum is used to describe how these particles absorb energy, in that they can only absorb discrete packets of energy to reach discrete levels of excitement. Quantum mechanics also deals with Symmetry of systems to describe how many Degeneracies there are. We say 2 states are Degenerate when they both have the exact same energy level. Some of you might have seen Quantum mechanics in Chemistry already. Atoms can hold an s orbital, 3 p orbitals, and the 5 d orbitals and so on. Those orbitals are symmetrically the same, and so they are energetically equivalent, i.e. degenerate. This applies to atoms only, the symmetry of some orbitals is Lost in molecules, sometimes the energy level is lowered (favorable), but it's anti-symmetric molecular orbital will in return be higher in energy. Any electrons occupying the higher energy Antibonding orbitals will interfere with the Bonding orbitals, making the end result a Non-bonding system (no bond formed, nothing happens). This is pretty much how Quantum ties up in Chemistry, and it helps tremendously in explaining and describing how molecules come to be, and can be used to even predict possible molecules.
