Hi, Intz. I'll try to be as clear as I can.
As you probably have seen during your studies, any system in Q.M. is described by a wavefunction, which in turn evolves according to the Schrödinger Equation. That evolution describes all the possible states the system will occupy, in a way.
The wavefunction that describes any system is frequently found in a superposition of those states. That is, the system is literally in all its possible states at the same time, with each state contributing a little bit to the final wave.
Collapse is the observation that, upon being measured, the system suddenly, instantly loses all the information contained in the superposition, and all possible states the system could occupy disappear, except for one state. That is called the collapse because the wavefunction, which perfectly described a multitude of states interacting with each other, seems to suddenly... well, collapse into the observed state.
However, that is a serious problem. Collapse isn't described or explained anywhere in the axioms of Q.M., and there has been found no equation or formula that explains when that happens, or even what a 'measurement' means so that we could understand what happens. Really, there is an appearance that the wavefunction has collapsed, while Quantum Mechanics itself says that that doesn't happen. So we need to find a way to explain collapse.
The Copenhagen Interpretation of Q.M., generally regarded as the most famous and/or common view, just states that collapse exists, and that the wavefunction isn't a true part of the world, but is rather just a mathematical model of what happens, and leaves it at that. The collapse is simply accepted, and incorporated into the calculations when needed.
Recently, however, research has been done on an event called 'quantum decoherence.' Quantum decoherence is what happens when the quantum system interacts with the environment in a thermodynamically irreversible way. When that happens, the environment becomes part of the wavefunction - i.e. you can no longer describe the system only as a wavefunction, you have to take the environment into account, so that now the it describes system+environment.
That system+environment wavefunction still has a superposition of states: each state represents one different measurement having been made. That is, in for each possible measurement of the original wavefunction, we have a system+environment state that describes not only the observable quantity that assumed that state, but also the measurement apparatus that measured such quantity.
So, let's suppose that at first we had an electron in a superposition of spin up and down. Before it is measured, the wavefunction describes perfectly the simple case where the electron has both possible spins at the same time, and they can even interact with each other. Now, suppose that you measure it and detect spin up. Suddenly, all possibility of spin down is thrown away, and the two "possible" states have apparently been replaced by a single "real" state. That is the wavefunction collapse. But what decoherence suggests is that the states haven't collapsed: they still exist. But since the wavefunction now has to describe both electron and you, there is a superposition of the states 'electron up and you saw electron up' and 'electron down and you saw electron down'.
That is, you have become part of the wavefunction, and you are now in a superposition of states. Each of the two versions of you has observed the apparent collapse, when in fact what happened was that you are now part of the system.
So, summarising: wavefunction collapse is the apparent event where the system, after being measured, apparently loses most of its possible states and assumes only one of them. Most interpretations other than the Copenhagen describe the apparent collapse as an epiphenomenon of decoherence, with its consequences having been described above.
