Electric fields and magnetic fields are usually grouped with electric current, what is the relation?

An electromagnetic field is the spatial representation of the electromagnetic force exerted by charged particles. An important concept of electromagnetism, this field represents all the components of the electromagnetic force applying to a charged particle moving in a Galilean frame of reference.

The electromagnetic field is indeed the composition of two vector fields that can be measured independently. Nevertheless these two entities are inseparable:

separation into magnetic and electrical components is only a point of view depending on the study reference;

the Maxwell equations governing the two electrical and magnetic components are coupled, so that any variation of one induces a variation of the other.

The behavior of electromagnetic fields is described conventionally by the Maxwell equations and more generally by quantum electrodynamics.

The most general way of defining the electromagnetic field is that of the electromagnetic tensor of the special relativity.