Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the submicroscopic scale.Have you finished reading that three times? It's a bit of a mouthful, so I will try to make it more clear. Quantum mechanics explains how atoms behave. Atoms make up most of what we see every day, so QM (as I will refer to it from now on) explains why things (matter, mostly) are the way they are.
QM is surprising. For centuries, it was thought that Newtonian physics (what you learned in high school) explained the world we see. In fact, Newtonian physics is a pretty good approximation that explains our world. But if you get down small enough, things start to become weird. Imagine something moving down a hill, for example: me when I'm rushing to class in the morning. It's all one continuous motion of moving-down-hill-ness. As I go down the hill, I gain speed because the energy I had just by being at the top of the hill is being turned in to "kinetic" energy. Electrons are not like this. They are more like someone leaping down stairs. The energy still changes, but it is done in stages, with each stage having a fixed distance between it and the previous one.
This brings us to why it is called Quantum mechanics in the first place. Think of our English word "quantity". Matter and energy comes in discrete quantities, or quanta. If you think about this for a bit you will realize how surprising this really is.
Next week, if you are interested, I will talk more about things of which you may have heard, such as "the uncertainty principle" and "Schrodinger's Cat".