Quantum mechanics generally does not predict the outcome of any measurement with certainty. Instead, it merely tells us what the probabilities of the outcomes are. This leads to the strange situation where measurements of a certain property done on two identical systems can give different answers. The question naturally arises whether there might be some deeper reality hidden beneath quantum mechanics, to be described by a more fundamental theory that can always predict the outcome of each measurement with certainty. An analogy exists with opinion polling: it is not that opinions are indefinite, but only if a reasonable sample of the population has been polled does one expect the poll results, as statistics, to be in line with the trend in the population at large.
In other words, quantum mechanics as it stands might be an incomplete description of reality. Some physicists maintain that underlying this level of indeterminacy there is an objective foundation. Such a theory is called a hidden variable theory.
Einstein, Podolsky and Rosen argued in 1935 that such a theory was not only possible, but in fact necessary, proposing the EPR Paradox as proof. In 1964, John Bell showed, through his famous Bell inequalities, that the kind of theory proposed by Einstein, Podolsky and Rosen made different experimental predictions than orthodox quantum mechanics. Experiment showed the orthodox account to be correct, and the hope for a so-called local hidden variable theory had to be abandoned.
It is sometimes suggested that hidden-variables theories have been ruled out by the Aspect experiment (see the EPR paradox). This is a misunderstanding of the experiment. What it did was to show that attempts to explain quantum phenomena cannot retain both the principle of reality and the principle of locality. The usual interpretation discards the principle of reality.
Hidden-variable theories, with their underlying determinism, must be non-local, maintaining the existence of instantaneous causal relations between physically separated entities. Non-local theories, i.e. theories that allow systems to interact over distances with speeds greater than the speed of light, were not ruled out. The best-known hidden-variable theory, the Bohmian mechanics, of the physicist and philosopher David Bohm, created in 1952, is a non-local hidden variable theory.
It is thought to be empirically equivalent to orthodox quantum mechanics. It still enjoys a modest popularity among physicists. What Bohm did was to distinguish between the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles. When you perform a double-slit experiment (see wave-particle duality), they go through one slit rather than the other. However, their choice of slit is not random but is governed by the guiding wave, resulting in the wave pattern that is observed.
Such a view contradicts the simple location of events in both classical atomism and relativity theory. It points to a more holistic view of the quantum world. Indeed Bohm himself stressed the holistic aspect of quantum theory in his later years, after his conversion from Marxism to theosophy.
The main weakness of Bohm's theory is that it looks contrived - which it is. It was deliberately designed to give predictions which are in all details identical to conventional quantum mechanics. His aim was not to make a serious counterproposal but simply to demonstrate that hidden-variables theories are indeed possible.
See also the many-worlds interpretation for another way of understanding the implications of quantum theory.