It sounds like your video card probably went kaput. The only way to tell for sure would be to get a different one and try it.
Make sure your monitor is looking for the signal on the right port (digital/analog, if applicable) and try removing and reseating the video card itself. You could also try a different cable just to be sure, though the cable being bad is unlikely.
Yes, I already made sure that the monitor was looking for the signal on the right port. I'll try to reset the video card and see if that does anything. When I opened up the case (while the computer was turned on), the fan on the video card was running. Is it possible for the gpu to fail but for the fan to keep on going?