If I could Em I'd be a millionaire!!
Unfortunately - it's incredibly complicated and if you read about it and, to quote my ex university professor, "if it makes any sense then you have completely missed the point..."
It's [very very basically!] based on two things.
WAVE/PARTICLE DUALITY
This is the theory that things aren't split into particles and waves - everything is actually a particle AND a wave, and therefore exhibits particle and wave-like properties. Normal everyday objects are so large that the wave-bit is unnoticible, and things like the photons of light are so small that we see them as waves.
QUANTA (ENERGY LEVELS).
It was believed that electrons could orbit an atom nucleaus at any distance from it - but it turns out that in can't. It's can only orbit at certain particular distances, or energy levels. The energy is 'quantised' which is where the quantum bit comes from.
It's all based on ridiculously complicated maths that I don't have enough natural life to appreciate properly!
Try here - they explain it all in reasonably simple terms - it's all split into the relevent sections - important ones are probably PLanck and Schrodinger.
http://www.spaceandmotion.com/Physics-Max-Planck.htm