According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random. That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if you place a single unstable atom in a controlled environment, you cannot predict how long it will take for the atom to decay—but only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities.
Historically, in physics, hidden variable theories were espoused by some physicists who argued that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system; i.e., that quantum mechanics is ultimately incorrect, and that a correct theory would provide descriptive categories to account for all observable behaviour and thus avoid any indeterminism. The existence of indeterminacy for some measurements is a characteristic of prevalent interpretations of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle.
Einstein, the most famous proponent of hidden variables, objected to the fundamentally probabilistic nature of quantum mechanics, and famously declared "I am convinced God does not play dice". Einstein, Podolsky, and Rosen argued that "elements of reality" (hidden variables) must be added to quantum mechanics to explain entanglement without action at a distance. Later, Bell's theorem would suggest (in the opinion of most physicists and contrary to Einstein's assertion) that local hidden variables are impossible, leaving only nonlocal hidden variable theories as potentially viable. The most famous nonlocal theory is de Broglie-Bohm theory.