Here is my humble understanding. A dynamical symmetry is a _hidden_ symmetry. The classic example would be the Hydrogen atom. Naively, we would only expect an SO(3) symmetry associated with rotational symmetry. This would be the geometrical symmetry, which leads to the conserved angular momentum vector. In fact, the full symmetry of the system is SO(4); this is exhibited by there being another conserved vector, the Laplace-Runge-Lenz (LRL) vector.
Since the LRL vector is peculiar to the particular potential of the hydrogen atom and does not emerge as the result of some general geometrical feature shared by a whole class of systems (like rotational symmetry), it is termed a _dynamical_ symmetry. If one were naively observing the Hydrogen atom, then one would only notice the extra symmetry in studying its dynamics.
Disclaimer: This is only what I have gleaned from reading some papers on dynamical symmetry; I have never read an actual definition.
* If I recall correctly, the SO(4) symmetry of the Hydrogen atom can be realized by starting in a four dimensional space and dimensionally reducing. In which case the dynamical symmetry starts out as a geometrical symmetry.
* Dynamical symmetry breaking is a type of spontaneous symmetry breaking and is an unrelated topic.