The thermocouple you have is what is called a pilot generator. It produces the voltage to operate your gas valve. It has to be producing 600 -700 millivolts d.c. for it to operate. Since this is way less than 1 volt, it needs to be in good tune to work.
It will look similar to this.
The first thing you need to do is look at the pilot flame itself.
It needs to envelope the entire end of the pilot generator.
Look for a good crisp blue flame. If you have a lazy yellow flame that curves upward and doesn't envelope the end of the generator, it will not heat it sufficiently to produce enough voltage to hold in the pilot valve.
If that is the case, a good cleaning of the pilot burner is in order. Use a small stiff brush, some folks have used a pipe cleaner and some compressed air to clean any dirt, soot , debris away from the pilot orifices.
This should take care of it. If not, you may have a faulty pilot generator. I would test the output of the current pilot generator before replacing it. It would require a decent digital volt meter capable of reading dc millivolts. If you are not able to test it, replacing it would be the next step. A faulty gas valve would be the last option, but fairly rare compared to the pilot issues discussed above. A fireplace shop or maybe even a decent hardware store should carry a new pilot generator.
Check the easy stuff first. Let me know if you have further questions. Thank you!