Electric current is the movement of electric charge through a conductor. For example, an electric charge carried by electrons through a wire.
An electromagnetic wave does not require a conductor. Electromagnetic waves are created by moving electric charges, but once created, they can propagate through a vacuum. Photons are the smallest packets of energy that propagate as electromagnetic waves.
In a vacuum, photons travel without being absorbed and re-emitted by a medium. But if an electromagnetic wave propagates through a medium, photons excite atoms, and this excitation is passed from atom to atom through the medium. The absorption and re-emission of photons slows the wave as it passes through the medium. Therefore the speed of an electromagnetic wave propagating through vacuum is greater than any propagation through a medium.
You can "feel" an electromagnetic wave if it excites the atoms in your body and causes them to vibrate as they absorb and re-emit photons, giving off heat. For example, sunburn is caused by photons from the Sun.
Electromagnetic waves should be distinguished from electromagnetic fields, which exert force upon electric charges in the vicinity. When an electromagnetic field oscillates, it manifests as the propagation of an electromagnetic wave. When it is static, it manifests only by its effect upon an electric charge.
There has been a great deal of controversy about the effects of electromagnetic fields upon humans. Electromagnetic hypersensitivity to electromagnetic fields has been reported, but no scientific basis for these reports has been found.