Perhaps the simplest natural form of radiation is Cyclotron radiation. If you have charged particles flying around in space, they would normally travel in straight lines. If, however, a magnetic field is present, they move in spirals around the magnetic field lines instead. All currents, in the presence of a magnetic field, experience a force perpendicular both to them and to the magnetic field. This then pushes the charges sideways, making them spiral around the field lines.
Synchrotron radiation is simply cyclotron with the particles moving at relativistic velocities.
A relativistic electron moves around the field lines just like any other electron. In its own frame of reference, it radiates just like a cyclotron electron. However, you have to convert from its frame of reference to that of an observed, using the Lorentz transform. Any light source that is moving relativistically beams most of its radiation into a tight forward cone. Thus we only see radiation from a synchrotron electron when it's path is directed straight at us.