The dictionary definition of light is electromagnetic radiation of any wavelength. Different types of lights are defined, and categorized by their wavelength in the electromagnetic spectrum.
 

The light that our eyes can see is only the smallest fraction of the electromagnetic spectrum (from 4,000  to about 7,700 angstroms) . We can see this light because it's wavelengths are detectable by the cones of our eyes. We can not see the other lights because they have too short of a wavelength, or to long of a wavelength.

Another uniqueness of synchrotron light is that it can be generated across the range of the electromagnetic spectrum including visible, infra-red, ultra-violet, 'soft' and 'hard' x-ray and microwaves (see the spectrum below). Synchrotron scientists are  trying to push synchrotron light farther into the X-ray and microwave regions and the brightness has improved over years as shown in the picture above.

The electromagnetic spectrum below shows wavelengths of radiation emitted by various sources and the names of the radiations based on their wavelength frequencies and sizes and lots more. You will learn a lot from this picture if you look carefully.
 

Synchrotron light is not just limited to Synchrotrons. As mentioned above,  the real name of Synchrotron light is electromagnetic radiation. Anything that emits any form of electromagnetic radiation from microwaves to x-rays is emitting a form of Synchrotron light. The only difference is that Synchrotrons makes the radiation up to a billion times more brilliant than the normal radiation emitted by other man-made objects.