In classical physics, there is no randomness. In quantum mechanics, there may be. Quantum mechanics is all just theory at this point, but it is commonly believed that quantum mechanical systems can exist in a superposition of states which then collapses (wavefunction collapse) into a single state. What state it collapses into is determined by probability; it is not necessarily guaranteed that it will collapse into a given state.
What causes wavefunctions to collapse is not known. It is generally said to be 'observation', but not everyone agrees on what 'observation' is. This is part of why there are multiple interpretations of quantum mechanics.
Of course, quantum mechanics in its entirety is just theory, so even the concept of wavefunctions and superpositions is just theory.