Most important application: produce X-rays for X-ray crystallography.
Note however that the big experiments at CERN, like the Large Hadron Collider, are also synchrotrons.
List of facilities: en.wikipedia.org/wiki/List_of_synchrotron_radiation_facilities
Predecessor to the synchrotron.
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A particle accelerator increases the kinetic energy of charged particles through the work of an electric field. A single electric potential difference gives an energy gain of magnitude to a particle of electric charge crossing in the accelerating direction. Reusing a smaller electric potential difference many times can give a much larger final kinetic energy without requiring the whole apparatus to withstand the equivalent single voltage. A magnetic field can return the particles to the accelerating region: magnetic forces do no work, but they make repeated energy gains possible. The cyclotron turns this separation between bending and acceleration into a particularly elegant machine.