How are binary star systems created? I don't know how common it is for a system to have two stars (or perhaps even more) but how do they arise? Is that due to the stellar accretion disc, or the composition of the stellar nebula? 
Or are most of them created by a collision of stars?
There are two main theories for the formation of binary stars - one accepted, and one mainly deprecated.


The fission hypothesis. The fission hypothesis states that the binary system forms after the collapse of the original gas cloud into a protostar. Angular momentum is conserved, so as the extremely large cloud slowly contracts, it spins faster. After enough time, the protostar could form a dumbbell shape, one part of which eventually breaks off. There are now two blobs, and thus two protostars, which will evolve into a binary star system.
The fragmentation hypothesis. The fragmentation hypothesis states that the gas cloud fragments earlier on, due to some instability or cooling/heating effects. Each of the two fragments then evolves separately, forming a binary star system.


The fission hypothesis is no longer favored. It cannot explain the existence of systems with certain mass ratios or separations. The fragmentation hypothesis, however, can, and is accepted. It is also possible for a binary system to form through the capture of one star by another, but that requires a third body, and probably cannot explain the high fraction of binary systems in the galaxy.

Basically, then, the main theory to date is that the protostellar nebula breaks up, and then both fragments collapse independently.