Quantum mechanics is, together with Einstein's relativity theory, one of the greatest intellectual achievements in physics of the last century. It is said to be the best explaining model of the reality physicists developed so far, sometimes quantum mechanics is postulated to be the reality, not just a model of it. Almost (if not) everything that surrounds us - a table, the Earth, the Sun, light, our bodies, aggregate states (like ice, liquid water, steam) is a manifestation of quantum mechanics.
Remarkably, the postulate that quantum mechanics is the reality (and not a just a model of it) stems from the fact that quantum mechanics predicts that the outcome of experiments will follow some strict rules, as strict as the mathematics behind. Another reason for this postulate is that these predictions were made long before they could be experimentally confirmed. The success of the predictions also created the question, why theoretical rules of mathematics are so successful in describing the world around us - in small and in macro scales.
Besides philosophical questions like these, what is special about quantum mechanics is that it is still a very vivid piece of physics, even after more then 100 years of research. It is also one of the most controversially discussed piece of physics, a reason why it is universally agreed to be not properly understood.
Quantum mechanics has still many unresolved problems, some of them theoretical in nature, some of them stemming from fundamentally mysterious experiments.
One example of such a mysterious experiment is the double-slit experiment, an experiment showing that a particle (say an photon) can fly through a wall with two slits passing both slits simultaneously, like it split into two parts passing both slits. However, when physicists try explain this behavior of the particle and observe it, the particle seems to "realize" being observed and passes only one particular slit, i.e. behaves "normally". Surprisingly, what is strict about quantum mechanics is that this experiment predicts always the same outcome - an observed particle will pass just one slit, an unobserved both slits simultaneously.
Another mysterious effect predicted (and seemingly contradicting Einstein's postulate that nothing can "travel" faster then light - about 300.000 km per second) is a quantum-mechanical effect called entanglement of particles. In this state, e.g. two particles can behave like they were connected by some mysterious connection - enabling them to exchange information about further changes of their state simultaneously. What is special about entanglement is that this information "travels" faster then light, in fact, at an unlimited speed - no matter, how far away from each other the particles are placed in the universe, they will interchange the information about their state just simultaneously (Einstein referred to this phenomenon as "spooky action at a distance").
What sounds like science-fiction, is observable in experiment and predicted by quantum mechanics - a remarkable piece of physics, to which this branch is dedicated.
