An odd aspect of Quantum Mechanics is contained in the Heisenberg Uncertainty Principle (HUP). The HUP can be stated in different ways, let me first talk in terms of momentum and position.
If there is a particle, such as an electron, moving through space, I can characterize its motion by telling you where it is (its position) and what its velocity is (more precisely, its momentum).
The preceding is a statement of the Heisenberg Uncertainty Principle. A consequence of the Uncertainty Principle is that if an object's position x is defined precisely then the momentum of the object will be only weakly constrained, and vice versa. One cannot simultaneously find both the position and momentum of an object to arbitrary accuracy.
We do not know if this indeterminism is actually the way the Universe works, because the theory of Quantum Mechanics is probably incomplete. That is, we do not know if the Universe actually behaves in a probabilistic manner (there are many possible paths a particle can follow and the observed path is chosen probabilistically) or if the Universe is deterministic in the sense that I could predict the path a particle will follow with 100 % certainty.
A consequence of the Quantum Mechanical nature of the world is that particles can appear in places where they have no right to be (from an ordinary, common sense [classical] point of view)! This has interesting consequences for nuclear fusion in stars.