Practically, how does an 'observer' collapse a wave function? I have been reading/learning about the double slit experiment, its implications in quantum theory, and how it explained that “particles” can behave as both waves and particles. 
I know that the wave function is a probability of the location of the particle, and that shooting the electrons through the double slits causes an interference pattern associated with multiple waves. This, though not making intuitive sense (in relation to how anything can even exist as a wave), is something I can follow. 
However, I have read/heard that an “observer” collapses the wave function into a single point. This is what caused the electrons to actually show up on the wall behind the slit; however Feynman (admittedly, as a thought experiment) suggested that putting an “observer” prior to the slits would cause the electrons to fly through as particles, and leave no interference pattern on the back wall. 
What is an “observer”? How and why would the electron “know” it is being observed and therefore cause it to change behavior?
 A: The other answers here, while technically correct, might not be presented at a level appropriate to your apparent background.
When the electron interacts with any other system in such a way that the other system's behavior depends on the electron's (e.g., it records one thing if the electron went left and another if it went right), then the electron no longer has a wave function of its own: the electron+"detector" system has a joint state. The two are entangled.
The electron doesn't have to "know" anything. The simple physical interaction results in a state vector which, by the laws of quantum mechanics, will preclude interference by any of the subsystems of this larger system. That said, the joint state can itself show a kind of "interference effect" (though not the kind you normally think of in the two-slit experiment).
If this entanglement is well-controlled (as in a lab), then (a) showing this "joint interference" might be practical, and (b) undoing the entanglement is also possible, thus restoring the electron's sole superposition. This is how we know that it hasn't "collapsed."
But if the entanglement is caused by stray photons, air molecules, etc., then any hope of controlling them becomes almost immediately dashed, and we can no longer exhibit interference in practice. From here on out, the system will appear to behave classically, with the different branches evolving independently. This fact is called decoherence. The superposition still hasn't "collapsed," but we no longer have the ability to show or exploit the superposition.
You may notice that this still leaves open a crucial question: when do the many branches become one? This is called the measurement problem, and physicists don't agree on the answer even today.
A: Wave function collapse only happens in the head of the physicist.
What we are dealing with is entanglement of the electron and the detector wavefunctions. In the double slit problem we can write the electron wave function as $\psi_L + \psi_R$. The detector has two orthogonal states, $L$ and $R$. If there is no detector we have interference. If there is one and if it distinguishes the two possibilities with 100% certainty then the wave function must be $\psi_LL + \psi_RR$. This is an entangled state where interference is absent as
 $\langle\psi_LL | \psi_RR\rangle$
$= \langle\psi_L | \psi_R\rangle\langle L|R\rangle =0$. 
No collapse occurs unless during the installation of the detector. 
A: Wavefunction collapse is a feature of the Copenhagen interpretation, which is one interpretation of quantum mechanics. It isn't the only one. These days people don't really talk about interpretations of quantum mechanics. They talk more in terms of decoherence. One of the things that was always unsatisfactory about the CI was that it never defined what was meant by terms like "observer" and "measurement."
A more natural way to think about this is in terms of decoherence. When a quantum-mechanical system interacts with an environment, there is a tendency for its phase information to get scrambled. Decoherence is a theory that allows us to calculate this sort of thing, and, e.g., find the time-scale on which this phase information is lost. When the environment is a big thing with a lot of energy, the time scale for decoherence is very short. When people talk about observers and measurement, they're talking about objects so big and containing so much energy that this time scale is much shorter than any other time scale in the problem, and therefore it makes sense to treat it as an instantaneous collapse, as in CI.
A: The early years of quantum theory were dominated by a school of thought known as the Copenhagen interpretation.
According to that school of thought the wave function of a particle could undergo an instantaneous change when some property of the particle was measured. The act of measurement was assumed to cause the change (which is sometimes called the 'collapse' of the wave function). So the short answer to your question, according to the Copenhagen school, is that an observer brings about a collapse of the wave function by making a measurement. For example, if a photon interacts with a photographic plate to produce a dark spot, the position of the photon is suddenly localised.
Many physicists have raised objections to this interpretation, those objections being on three main grounds. Firstly, the collapse seems to be instantaneous, with no supporting theory about what mediates or triggers it. Secondly, a 'measurement' is just an interaction between the particle and some other particle that happens to be part of the measuring apparatus. And thirdly, that the measuring apparatus itself is just a collection of particles with wave functions, so why should it not be subject to the same sort of discontinuous change along with the object it is measuring?
These objections still haven't been been fully resolved. Many resolutions have been proposed, and each has its proponents and detractors.    
A: A photon is, or is associated with, an electromagnetic wave packet.  Its energy can be thought of as being embodied in the energy density of the electric and magnetic fields.  A wave function describes this wave packet.  Observing a photon generally means that it has been captured (as in a CCD or on a piece of film). In being captured, the photon looses its energy to the capturing device and the wave disappears.  There is nothing left for the function to describe.
A: Physics students are taught the following three things:
1) A wavefunction is a probability density function, infinite in size, which serves as a useful fiction, allowing us to calculate properties of a particle.
2) Wavefunction collapse is a real event, a non-fictional event, initiated by something external to the particular wavefunction in question.
3) Physicists have some idea what's going on in the universe, so we should take them seriously.
Clearly, one of these three things needs to go.
