I'm coming to the end of my second year in maths/physics with a quantum exam coming up soon, and I've got a loooong way to go before I feel confident in saying I understand the stuff.
Basically though, everything in the universe is quantised (hence the name), distance, position, momentum, energy, charge, etc.. This means that individual particles can only fall into discrete quantum states, the simplest version of this would be electron shells in an hydrogen atom. The electron can only take certain energy and momentum values, leading it to only exist orbiting at a discrete distance from the proton.
The two other majorly mind boggling things are wave-particle duality and the uncertainty principle. Wave particle duality stats that everything is both a wave and a particle, and will behave as one or the other simultaneously, sort of.
The uncertainty principle says that for certain values, the more precisely you know one, the less precisely you know the other. For instance you are allowed to know the total angular momentum of a particle at the same time as one component of the angular momentum ( taken as the z component) at the same time. But knowing both the x and y components a the same time is forbidden. Momentum and position have a similar relationship.
One of the principles of quantum mechanics is that as the quantum numbers are scaled up, the way particles behave becomes much more normal until it fits a classical model of physics. Think of a digital sound signal looking like an analogue wave at low resolution.
These three things more or less govern most quantum interactions as far as I'm aware. Lots more things are massively confusing if your not used to pure mathematics (such as the order of addition effecting the result mention somewhere earlier). I'd explain more, but I should get back to revision
