Electricity (more precisely electric field) and magnet (let's say magnetic field) are two sides of the same coin. In modern physics, electric and magnetic field are written in the same equation. Changing electric field (time-dependent electric field) will generate magnetic field, and changing magnetic field (time-dependent magnetic field) will generate electric field.

Understanding the relation between electric and magnetic field was the motivation for Einstein to invent relativity. His first paper on this subject was "ON THE ELECTRODYNAMICS OF MOVING BODIES".

To see the effect of electric field on magnet, you can put a compass next to an electric cable. When you turn on and off the electric current, the compass will change its alignment.

Electrical currents create magnetic fields as a consequence of the changing (flowing) electrical field responsible for the electricity. Anything that results in a changing electrical field will create a magnetic field and vice-versa.