Quantum science, also called quantum mechanics, is the branch of science that describes the behavior of photons and sub-atomic particles. This behavior is very different from what is observed in the world of classical physics. The word quantum describes something that only exists in specific amounts. A staircase is an analogy of something that is quantized, since you can only stand on one step or another, not in between. An elevator would not be quantized, as it's not limited to certain allowable positions.
Electrons and light are both quantized. Electrons can only have certain specific energies and light exists in photons which are discrete packets. The particle-wave duality of light and electrons is at the heart of quantum science. The wave function, developed by Erwin Schrodinger, describes the position, momentum and other properties of a wave/particle via a complex mathematical equation. These properties and behaviors are quantized, which allows unusual behaviors such as simultaneous motion in more than one direction. A wave function with two solutions allows two possibilities for the property in question.
MRIs, electron microscopes, super-conducting magnets and semi-conductors arose from quantum science. The idea of parallel universes also comes from quantum science.
If you're interested in reading a good book on the topic, I recommend Warped Passages by Lisa Randall. She's a Harvard physicist with a very engaging writing style, and she begins each chapter with a popular song lyric :-)