The argument, at least in its modern form, focuses on two primary domains. Firstly, the complexity of living organisms and their origin, and secondly the origin of our universe. It is no coincidence that these two areas are both at the boundary of modern scientific understanding. You can trace religious apologetic arguments right back through time and notice how these arguments always ride on the boundary of the scientific understanding of the day, and fall out of favour as that boundary progresses outwards (i.e. as scientific understanding increases).
The argument from complexity is really a separate argument in its own right, and with its own flaws, but the way I approach it is more or less the same as with the various Fine Tuning arguments. I understand why these arguments are compelling to people, but I think they are compelling primarily due to subtle flaws or mistakes in our reasoning process.