The fan on my heat-sink died and I need to know how to tell what type of heat-sink it is, if there's anything beyond thermal gel that I should think about?
The fan/heatsink combo is normally designed to fit a certain type of processor architecture. For example, an Athlon 64 processor may fit a Socket AM2 motherboard, and thus you would need a fan/heatsink combo that is compatible with a Socket AM2 motherboard. Also, sometimes the socket/slot type is printed on the motherboard.
Most fans are specially made for a certain heatsink, but it not always the case. As such (and as prices for fan/heatsink combos are relatively cheap) it is best to look for a new fan/heatsink combo.
Be wary of branded PC's and replacement fan/heatsinks. Sometimes the new features of a fan/heatsink will not work with a branded PC. I recently had an HP PC that had hardcoded the fan speed minimum into the BIOS--when I tried a new fan/heatsink that didn't need to run as fast the BIOS thought the fan was broken and shut down the PC. If it's a branded PC I'd get the part number from the Manufacturer and start there.
On that same note, custom PCs are designed to be compatible with a plethora of fan/heatsink combos.