The collapse of the wave function by measurements is one of the most mysterious properties of quantum mechanics.
At what scale does the wave function collapse? What are the conditions for a collapse?
Can it be made precise?
Does the collapse happen instantaneously, or is it spread out in time?
What is the precise equation for the collapse?
Why is the collapse time-asymmetric, and can this explain the arrow of time?
Dear Jack, there is no physical phenomenon that could be called the collapse. The collapse of the wave function, as first emphasized by Werner Heisenberg and then many others, is just the event when we learn something about a physical property of a physical system. When we learn that Osama bin Laden is located in a building in Pakistan, his wave function - that could have included possible positions at many other places - suddenly "collapses" because we learned about the position. That was pretty much Heisenberg's description of the situation.
The wave function is not an actual wave - like an electromagnetic wave. It is a collection of numbers that summarizes our knowledge about the physical system and that can be used to make predictions. Any attempt to "overinterpret" the wave function and "visualize" it as a real wave that objectively exists etc. is fundamentally flawed. The collapse of the wave function is just a process in our brain when we learn that the physical quantity $A$ has taken one of its values, $A=A_i$. When we learn it, we must replace the calculation of probabilities such as $P(B=B_j)$ by the calculation of the conditional probabilities such as $P(B=B_j|A=A_i)$. This replacement may be visualized as a collapse but no actual collapse occurs - and by this statement, I mean that all questions about the detailed "mechanism" of the collapse are based on an incorrect assumption that there is a mechanism.
Another question is at what sense it's possible to say that $A=A_i$ was "perceived" by "something" or "somebody". Quantum mechanics surely prevents one from declaring such a measurement too early. You never make an error if you continue to use the whole wave function and only "collapse" it in your mind when you want to learn some particular answers. So you may evolve the electrons, hammers, cats, and other people according to Schrödinger's equations so that the wave function spreads into many possibilities, and only when you know what you feel, you may make the "collapse" in your brain according to what you felt, and continue with the simplified wave function.
Alternatively, to simplify your thinking, you may imagine that the wave function for the other objects collapsed earlier than that - you're allowed to imagine that there was a collapse after a sufficiently "macroscopic" interaction. What it exactly means? It means an interaction in which the different values of various properties have "decohered" from each other. Decoherence is a derivation of the classical-quantum boundary - of the point at which it becomes pretty much legitimate to imagine that the objects have well-defined properties such as positions, just like in classical physics.
But one doesn't need to imagine that this reduction ever takes place.
Contrary to the general tone of the answers I have read so far, I feel there are indeed real difficulties in quantum mechanics which fall under the general category of "collapse of the wave function". I think part of the problem is the tendency of people to talk in generalities instead of dealing with specific issues. I am therefore going to list a few of what I would consider some of the more glaring instances of wave function collapse.
The flecks of silver on a photographic plate when exposed to the light of a distant star. Whether the wave function is "real" or simply a form of information, by any conceivable calculation it is awfully spread out by the time it reaches the photographic plate. How does the energy concentrate itself enough to activate a highly localized chemical transition?
Radioactive decay. As far as I know, according to the equations there is a superposition of uranium and lead, or radium and radon, and based on this superposition one can calculate that there is a slow but steady emission of gamma waves and alpha waves. The problem is the clicks in the geiger counter. You don't need to hook up a detonator to a box with a cat in it to see that there is a problem here. (There is also a description of the emission whereby it is a tunneling process of alpha particles trapped in the potential well of the nucleus; I think this is equivalent to the way I described it except for a change of basis states, but if I am incorrect it nevertheless remains the case that there is a wave function describing the slow but steady emission process.)
The tracks in a could chamber. The emission of charged particles from a decaying nucleus, according to almost any kind of wave equation, ought to be if not spherically symmetric then at least some superposition of spherical harmonics. Yet the observed tracks in a cloud chamber are alwyas straight as an arrow.
Jack asks several questions which I cannot answer, except to some degree where he asks if the process is instantaneous or spread out in time. The problem of course is that the collapse seems to be instantaneous. A mechanism whereby the process can be explained through normal time-evolution of the wave function would make the notion of collapse unnecessary. It is my personal opinion that eventually such explanations will be found.
There are currently two different accounts that give a larger picture of what happens when a quantum system is measured.
One of them is the fact that many random interactions between the system (which might be a 1-body or N-body quantum system) and the environment (which is considered for most purposes a pseudo-classical system with infinite degrees of freedom) produce interactions that tend to average out interference terms (which are represented by diagonals in the density matrix), leaving only a pure diagonal matrix. the pure diagonal matrix is what one would use to describe a classical system with a number of classical probabilities of being in several states.
The other one is based on the idea that once you take into account the should-be-obvious fact that any observer of quantum systems must be described by the same quantum laws as any other physical system, becomes clear that the phenomena of collapse is not an ad hoc postulate you need to plug into the theory by hand, but it is the unescapable consequence of interaction between (quantum) observers and quantum systems; any time a quantum system in superposition of states interacts with a quantum observer, the whole system + observer will evolve into a superposition of entangled states between each eigenstate in the original superposition of the system $\times$ a new observer state where the observer is aware of a different eigenvalue of the system he perceives to have measured by the interaction
An often good on-line source for the interpretation of quantum theory is the Stanford Encyclopedia of Philosophy, which has a page on "collapse theories".
There is a lot of literature on whether one needs collapse if one takes the wave function seriously, as opposed to the mainline Physicist's approach of taking a more empiricist view, as outlined well by Luboš, but the non-collapse alternative is epitomized for most people by the many worlds interpretation.
