Classical electrodynamics is the discipline that studies electromagnetic phenomena – such as electric and magnetic fields, radiation, and the dynamics of charged bodies – in classical terms.
When to Use this tag.
electromagnetism covers the classical description of both static and dynamic electromagnetic phenomena. classical-electrodynamics is typically used, as an alternative to electromagnetism, to emphasise that the question focuses on the dynamical aspects of electric and magnetic phenomena, as opposed to electrostatics and magnetostatics. On the other hand, the tag quantum-electrodynamics is used for the quantum-field-theoretic approach to electromagnetism.
Background.
The basic equations of classical electromagnetism are the Maxwell equations, $$ \nabla\cdot\vec E=4\pi\rho$$$$ \nabla\cdot\vec B=0$$$$ \nabla\times\vec E=-\frac{1}{c}\frac{\partial\vec B}{\partial t}$$$$ \nabla\times\vec B=\frac{1}{c}\left(4\pi\vec j+\frac{\partial\vec E}{\partial t}\right) $$ together with the Lorentz force, $$ m\dot{\vec v}=q(\vec E+\vec v\times\vec B) $$
The first set of equations, together with some boundary conditions, determine the electric and magnetic fields uniquely. Furthermore, given the pair $\vec E,\vec B$, the last equation, together with some initial conditions, determines the position and velocity of the point particle uniquely. Thus, the Maxwell equations together with the Lorentz force are enough to describe all electric and magnetic phenomena, and their effect on charged bodies.
As it turns out, classical electromagnetism is in fact Lorentz covariant, although this is not obvious from the formulation above. Introducing the so-called field strength tensor $F_{\mu\nu}$, and the four-velocity $u^\mu$, the equations above can be recast in a manifestly covariant form, to wit, $$ \partial_\mu F^{\mu\nu}=j^\nu,\qquad mu^\mu=qF^{\mu\nu}u_\nu $$
For more details, see electromagnetism.
