"Central Limit Theorem" says that "adding (bounded variance) stuff together makes it converge to a Gaussian" (roughly). On the other hand, when you are adding random variable together, you are actually convolving their densities. Thus, what the CLT says is that a gaussian is some sort of attractor/fixed point of the "dynamic process" of convolving (finite energy/bounded variance) distributions.
There are (several) central limit theorems for dependent variables, but you often have to assume other things as well (e.g., stationarity, bounded third moment, and/or limited-range correlations) for it to work.