Dynamics is the study of change, and a Dynamical System is just a recipe for saying how a system of variables interacts and changes with time. For example, we might want to understand how an ecology of species interacts and evolves in time so we can answer questions like, ``how robust is this system to small changes'' or ``if we decrease the rainfall by 10% or make it erratic, will the system crash and burn? or will some species flourish''. Similar questions can be asked for the economy, the stock market (they may not be the same thing), simplified climate models, or reactive or radioactive chemicals in groundwater. The different systems may seem to be distinct, but they can often be investigated using the same powerful tools.
When we speak of dynamical systems mathematically, we are talking about a system of equations that describe how each variable (e.g. each species) changes with time.
The n species are given by ( ) and the right hand side of each equation is a function that says how fast that variable changes with time. In general, the rates of change will depend on the values of the other variables and this is what makes the business interesting. If they depend on each other in a nonlinear way, then things can get really interesting. Nevertheless, the important point is that as long as we can evaluate the different functions for a given set of variables and time, we can always say something about how the system will evolve. We will use this trick extensively, to show that you can often understand the behavior of the entire systems (sometimes) without even solving the differential equations. However, at this point, things are a bit too abstract so lets start from the very beginning.