Light waves result in there being electric fields and magnetic fields in parts of space. Where the electric and magnetic field from one wave are the opposite sign but the same strength as another wave they cancel so there is no electric field. If the waves are of different frequency this only happens for an instant, at some places, but if the two waves are "coherent", for example if they both come from one laser beam that has been split, then there can be some places where the electric and magnetic fields cancel all the time. If this happens, there will be somewhere else where the fields are double, and the total power passing at any point is always equal to the sum of the powers passing in the original two beams. You may ask what happens if the two beams are exactly parallel and on top of each other so that they seem to be cancelling everywhere? In this case either (a) the source producing the two beams sees their interaction as an impedance change and produces no power or (b) in something like a beam splitter/combiner (which would be needed to produce this), all the power went in another direction.
Light waves result in there being electric fields and magnetic fields in parts of space. Where the electric and magnetic field from one wave are the opposite sign but the same strength as another wave they cancel so there is no electric field. If the waves are of different frequency this only happens for an instant, at some places, but if the two waves are "coherent", for example if they both come from one laser beam that has been split, then there can be some places where the electric and magnetic fields cancel all the time. If this happens, there will be somewhere else where the fields are double, and the total power passing at any point is always equal to the sum of the powers passing in the original two beams. You may ask what happens if the two beams are exactly parallel and on top of each other so that they seem to be cancelling everywhere? In this case either (a) the source producing the two beams sees their interaction as an impedance change and produces no power or (b) in something like a beam splitter/combiner (which would be needed to produce this), all the power went in another direction.
The stonishing answer of Malcolm is sufficiently complete. But there something which is required to be noted. In any Linear Time-invarant system, just like much of phenomenon around us, the Principal of Causality is held. so the Superposition effect can be foreseen. The effect which you asked about is always happening around us. At any arbitrary point of space the final intensity of visible light( or any electromagnetic wave) is equal to algebraic sum of all passing waves at specific time at the point. but in special cases you can see the effect as described by Malcolm.
Malcom's answer is excellent. I would add that in the quantum mechanical realm, the wave amplitude loosely corresponds to the probability function that a photon will be in a certain place. Where the probability wave cancels, there is no possibility of a photon being there (but the probability elsewhere increases). When large numbers of photons are present, this amounts to the same thing as the field description.