Originally magnetism and electrodynamics were believed to be separate, but related phenomena.
However, Electromagnetism is the word for the combined theory of electrostatics, magnetostatics and electrodynamics and magnetodynamics that effectively states that they are all different sides of the same coin.
The convergence occurs partly because of Maxwell's Equations which shows how they interelate for a fixed frame of reference.
However, that they are the same thing is most elegantly described by Special Relativity; which shows how simple relative motion of a frame of reference can convert an electrostatic field into a magnetic field (and vice versa). In fact they form what is known as a four vector.
Due to Relativity some observers can correctly state that an effect is caused by an electrostatic field, whilst others will think the same effect is magnetic in origin, but all agree on what actually transpired. And thus this shows that they are the self same thing.