According to Revonsuo and Newman (1999) the binding problem is "…the problem of how the unity of conscious perception is brought about by the distributed activities of the central nervous system."
For instance, say that Jon sees a red balloon floating across a room. The quality of "redness" is said to be processed in one part of Jon's brain, the shape of the balloon in another, the size in another, and the movement in yet another. Where then, and how, are these qualities or qualia combined to form the unified experience ("red-balloon-floating") in Jon's consciousness?
A more complex example of the Problem involves the combination of data-sets created via multiple types (modalities) of sensory input, eg olfactory, visual and auditory (eg as in looking at a person, hearing them speak, smelling their body odour and chunking it all up into one discreet piece or lump of conscious awareness of that person).
An even more complex example relates to memory. How do we access an integrated tightly bound-up element of consciousness relating to the past, ie not happening now? How do we access a discreet element of consciousness consisting of multiple data sets derived in the past from multiple inputs from multiple and different sense perceptions? And what about imagination (the combination of data sets relating to things/events that may not yet have occurred in the “real world” and/or may never occur)?
In my view the Binding Problem is not a Problem but rather an opportunity. What does the process of binding up bits and pieces into one holistic lump remind you of? Creating reality, that's what it reminds me of: the new age idea that a person creates their own reality. Isn't that exactly what the “binding” part of the Binding Problem is: a person taking bits and pieces of primary and secondary data and binding them all up together to form a unified consciousness of reality?
It also reminds me of what quantum physicists believe about the role of the observer in determining the nature of reality, that is that observers create reality. For example, light comes in the form of particles, photons, or in the form of waves, depending on the experiment you conduct to determine the nature of light.
Quantum physicists believe reality can never be fully known.Heisenberg's Uncertainty Principle dictates that some data is gained only at the cost of making other data unattainable. Physicists believe that at the quantum level, things have the potential to be different things at once---a wave and a particle, here and there, now and then, up and down, and all around, etc. They believe a cat can be alive and dead at the same time.
They believe the act of conducting an experiment (ie observing) actuates or crystallizes the cloudy confusion of quantum potential into the superficially sure, steady, solid reality we all know and love, where a cat is alive or dead but not both at the same time. (In fact, Einstein’s theories of relativity demonstrate there is no such thing as the “same time”). Physicists interpret the act of observation as "collapsing the wave-function". From stochasticity to specificity, you might say, if you were being a dickhead. Or from stochasticity to casuistry. Or as the ancient Greek philosopher Heraclitus said, "From out of all the many particulars comes oneness, and out of oneness come all the many particulars."
So the Binding Problem is a problem because we don't know how the unity of conscious perception happens. So it's a problem for us because we don't understand it. In fact, not only do we not know how "it" happens, we don’t even know what "it" is, why “it” is, or even whether “it” happens at all.
There is still no good theory of consciousness: what it is, where it lives, whether it is in fact unified, etc. And there is still no good theory of personal selfhood: what it is, where it lives, whether it is one or many, where the boundary of the self is, etc.
There is no homunculus in the control room inside your skull behind your eyes. There is no control room, there is no ONE in control. Consciousness emerges when a certain level of complexity is reached -- it is susceptible to reductionist analysis only up to a point, and no further. And beyond that point, there is still a fair way to go.
Revonsuo, A and Newman, J. (1999). Binding and Consciousness. Consciousness and Cognition 8, 123-127.