Quantum mechanics is a mathematical framework for understanding physical phenomena that fail to be described sufficiently by classical physics: the work of Newton, Maxwell, and others. You might ask, 'Why can't we use classical methods at that point?' We're unable to sufficiently probe atoms and subatomic particles with classical instruments at that scale. The simplest instruments that we can use at that scale, like probing with an electron or photon, are either too energetic or don't have the right properties to measure what we want to measure. This doesn't just mean that WE are unable to measure it, but that, and this is what makes QM weird, physics doesn't allow us to learn any more about it. So instead of being able to watch particles zooming and blipping and popping about like we are able to watch a ball on a playground, we have to use statistical methods to determine where some particles might be and where it might be going. This is the very broad description of how QM works.
Wikipedia has a good historical intro which will provide you with who discovered what, when, and where, and how QM came to be essential for furthering our knowledge about the small-scale stuff. Unfortunately I lost my old account and can't post a link for you until I reach 10 posts again. But google 'intro to quantum mechanics' and it should take you straight there.