OK, this is a LOT to think think about! By aperiodic we mean that "the same state is never repeated twice". Actually, it really means that the states are definitely NOT periodic. For example, some chemical reactions are known to be periodic, so these dynamics could not be chaotic. For another, more mathematical example, click here. On the other hand, water never pours out in a "periodic way" out of a faucet, so this could be considered "aperiodic". Click here for another example.
Bounded dynamics refer to the fact that the system that we are studying is finite, or nothing in it gets "too big" (this is almost a tautology for all physico-natural phenomena). For instance, the function whose graph is given by this link is bounded whereas the function whose graph is this one is not bounded.
Now the notion of sensitive dependence on initial conditions means that small changes in the way we start off our dynamics may produce "large changes" in the output, more specifically, exponentially large changes. We'll see a lot of this later on...
Basically, however, think of a chaotic system as one where predictability of future dynamics is inherently (almost) impossible. Think of the weather ... One of the big problems in this area is the question of whether or not our own solar system exhibits "sensitive dependence"; for example, what would happen if a small black hole crossed the plane of our solar system?? Would the planets "eventually resume" their elliptical orbits or would the result be catastrophic for life on our own planet?
