What exactly causes quantum wave function collapse? I've heard many things, such as the wave function collapses when the particle or system it determines is observed. But then I've heard that if we consider that to be the case, it's incomplete. What truly does cause wave functions to collapse? Maybe consciousness is one of many things that can cause wave functions to collapse? What component of observation causes collapse?
 A: Nobody knows. In large part, this issue and question have been swept under the rug for most of the twentieth century physics. If you have ever heard the nostrum of "shut up and calculate" as applied to Quantum Mechanics, you can safely assume that you are being instructed not to ask questions like that. 
What is more, there is no such thing as a "collapse" of a wavefunction. This idea of collapse was simply concocted to explain how one can go from what seems like a superposition of classically comprehensible states to something (a state) with a certain and classically measurable value.
Now, the question is, of course, a fascinating one; you will not find an answer in any of the standard QM texts. There are however theories that attempt to give more or less complete answers to this question. One of such theories (more of an interpretation, actually) that I am aware of is DeBroglie-Bohm theory. 
https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory
A: To add to ArtforLife's answer you are speaking of the famous Quantum Measurement Problem. I disagree with his/her answer a little (even though I upvoted it) insofar that we're not sweeping the question under the rug so much as using abstraction to decouple descriptions from one another so that we can tackle and conceive of one description at a time. In other words, Physics is very aware of the depth and importance of the question and it is an open and very active problem, but one can make considerable headway in theoretical and experimental physics without answering it through the appropriate abstraction. Namely, we postulate the mathematical idea of an observable to model this measurement process, which is an Hermitian operator together with a recipe for how this operator corresponds to the measurement process: the quantum state on measurement somehow winds up in one of the operator's eigenstates straight after the measurement; which one is random but the probability distribution describing the statistics of which eigenstate arises from the measurement is calculated through Born's probability interpretation. Notice how through this abstraction we decouple our outcome statistics calculation from the measurement problem. It is a simple "black box" kind of model, and the question of whether it is owing to anything as complicated (and as poorly understood) as "consciousness" is put aside for the calculation.
As to what goes on at "measurement", there are several current theories people are thinking about. There are a couple of posts on this site about how entanglement of the measured process with the quantum state of the measurement apparatus can give effects that are very much like the "discontinuous" state change that an observable-modelled measurement begets. See:
DanielSank's answer to the question "Really how can an observable quantity be equal to an operator?"
Lurscher's answer to the question "Is quantum entanglement functionally equivalent to a measurement?"
These ideas are further formalized and studied in the theories of Einselection and Quantum Darwinism. Essentially all these ideas fall under the general explanation that measurement may simply be a loss of knowledge that goes with the interaction between a known quantum state and an incompletely characterized, complicated measurement system.
I give the ideas above not so much to imply that the "measurement problem" is on the brink of being solved, but rather that (1) a solution is indeed plausible within even my lifetime and (2) that the problem is given ongoing serious thought rather than "swept under the rug".
