Quantum physics is, by nature, counterinuitive and illogical, so there is no easy way to fully explain the concepts that underpin it. Some of the more famous concepts are wave-particle duality, superposition and the uncertainty principle.
Wave-particle duality asserts that at the sub-atomic level, matter has properties of both particles and waves. In other words, it can effectively exist in multiple locations at the same time, with limited probability of being in one exact spot at a given time. This is the idea behind the electron cloud; it supplanted the idea of atoms being a miniature solar system.
Superposition is the idea that matter can exist in multiple states simultaneously, dependent on observation. The Schrodinger's Cat thought experiment examines this and was originally intended to refute the idea of superposition. In the experiment, a cat is placed in a box with a radioactive isotope, a Geiger counter and a vial of poison rigged to shatter if the Geiger counter is activated. The isotope has a 50 percent chance of decaying, triggering the Geiger counter and thereby killing the cat. Until the box is opened, the cat is simultaneously alive and dead.
The uncertainty principle holds that either a particle's velocity or location can be observed, but not both. In this instance, observation means interacting with the particle in question in any way.