The idea is similar to the role that atoms played in understanding thermodynamics. In the late 19th century, Boltzman proposed that heat could be understood as simply the kinetic energy associated with atoms, however, many senior physicists of the day disbelieved the idea that atoms existed. Einstein later described Brownian motion in terms of atoms bouncing off of dust, and 10 years later the idea of atoms became firmly established.
In 1932, the great mathematician John von Neumann wrote a highly influential book on Quantum Mechanics in which this theory was treated as a purely mathematical theory as though it were a branch of mathematics. He presented in this great work, a proof that no hidden-variable theory could ever reproduce the results of quantum mechanics. This is where the discussion remained until David Bohm, then in Brazil in the 1950's, refuted von Neumann's proof and wrote two papers which presented a specific model in which hidden-variables could exist, and in which quantum mechanics as we know it was preserved. However, each individual system is in a precisely definable state determined by definite laws. Quantum probabilities are a practical necessity, not a reflection that there is a lack of complete determination of the properties of matter. In other words, quantum mechanics was just another form of classical mechanics free of probabilities. indeterminism and all the other enigmas of the quantum world.
What Bohm had done is to find a statement by von Neumann that was true most of the time, but that under certain circumstances would not hold. This mathematical statement was the crux of his proof that hidden-variable theory was impossible. Bohm found an exception to this statement, and developed his model of a hidden-variable theory to occupy this logical niche in von Neumann's otherwise correct proof.
In the early 1960's the physicist John Stewart Bell and his physicist wife went to work at Stanford University. John Bell had always been intrigued and even a bit obsessed by the foundations of quantum theory, von Neumann's work, and the so-called Einstein-Podolsky-Rosen experiment, and he took this new opportunity to investigate this hazy area in physics. What he ultimately came up with was a surprisingly simple experimental test which defined in rather absolute terms just what kind of theory quantum mechanics is, and what the possibilities would have to be for ANY challenger to it.
Bell's Theorem, expressed in a simple equation called an 'inequality', could be put to a direct test. It is a reflection of the fact that no signal containing any information can travel faster than the speed of light. This means that if hidden-variables theory exists to make quantum mechanics a deterministic theory, the information contained in these 'variables' cannot be transmitted faster than light. This is what physicists call a 'local' theory. John Bell discovered that, in order for Bohm's hidden-variable theory to work, it would have to be very badly 'non-local' meaning that it would have to allow for information to travel faster then the speed of light. This means that, if we accept hidden-variable theory to clean up quantum mechanics because we have decided that we no longer like the idea of assigning probabilities to events at the atomic scale, we would have to give up special relativity. This is an unsatisfactory bargain.