I installed a new over the vanity light in my bathroom. After merging the wires and finishing installing I turned the breaker back on and flipped the switch, when I did this the light popped and now has a burn mark in the holder. What do I do now? Should I buy a new light, try to rewire it, or is the whole circuit gone?

No, yews and no. You don't need a new light, you do need to correct your wiring error and the whole circuit is not gone. The burn mark is a short where your hot black or white netrual is touching the base. Shut of the breaker and disassemble inspecting acrefully for where you oinched a wire.