The simplest way to make these fields is with electric charges (for an electric field) or with magnets (for a magnetic field). Charges are naturally surrounded by electric fields and magnets are naturally surrounded by magnetic fields. But fields themselves can create other fields by changing with time. That's how the fields in a light wave work—the electric field in the light wave changes with time and creates the magnetic field and the magnetic field changes with time and creates the electric field. This team of fields can travel through space without any charge or magnets nearby.
