What's the difference between cyclotron radiation and synchrotron radiation? The title says it. If there is a difference between a cyclotron and a synchrotron beyond the particles being accelerated in bunches, please clarify. Is bremsstrahlung a distinct mechanism from the above? Is there a gravitational equivalent, such as a relativistic particle moving just beyond an event horizon (or whatever the equivalent is since Stephen Hawking said something about black holes, the details of which I have no knowledge, maybe they were grey this whole time)?
Cyclotron radiation is the radiation emitted by a non-relativistic charge when it is accelerated by magnetic field. Synchrotron is similar for a relativistic charge with relativistic beaming and characteristic frequency approximately $\gamma^2$ times the cyclotron frequency. Bremsstrahlung is the radiation emitted when a charge is accelerated as it approaches a different charged object, often a nucleus. In all three cases, we use Larmor's formula for the power emitted.
A relativistic particle experiences a force similar to the Lorentz force but of relativistic origin, the one that causes the Lense-Thirring effect, which is stronger near a massive object. Your description about a particle near the event horizon indicates that you're probably referring to Hawking radiation, which is due to quantum effects.
I couldn't add the comment, but this is an excellent introduction to cyclotron and cyclotron radiation for future readers (along with explanations of relativistic beaming and other associated effects): https://www.youtube.com/watch?v=FnYNjKabmxU&feature=youtu.be
To reiterate what's been already posted: synchrotron emission is simply cyclotron radiation in the relativistic limit and thus there are additional considerations that need to be taken into account such as Doppler shifting.