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.