This morning the light bulb in my ceiling fan blew out with a rather loud "pop" sound. I replaced the bulb and nothing. I replaced the lampholder, bypassing the pull chain switch and nothing. The fan is working, so I have power. There is a small plastic box in the wiring to the lampholder with a trademark name of Vim and some electrical spec information printed on it. I don't know what that is. Is that where the problem lies?

No.

Is this a remote fan?

General ceiling fan troubleshooting and help FAQ- Ceiling Fans N More

No, this is not a remote control fan. Both the fan and the light have pull chain switches.

You basically have to trace the wire from the power source to the socket and back. Do you have a volt meter?

Are the fan and light on separate wall switches?

Pictures might help.

What's printed on the plastic box?

When bulbs blow, the filaments normally open. Occaisionally a piece of the filament will short. This can take out any protection device that may have been installed in the fan (plastic box?). The bulb could have also broke the switch.

In a car I had, the headlamps started blinking. The bulb shorted. The thermal fuse popped. It self reset. Lamp on, lamp off etc.

So, markings and/or picture might help.

The plastic box is most likely the fan motor capacitor.