Why does the pipe to my radiator get hot, but the radiator itself doesn't?
Do you know for sure that this is a hot water system? Sometimes steam can sound like running water. If it is hot water, there is a giant air bubble in the radiator preventing it from filling. It needs to be bled off by opening the small valve near the top at one end. You will typically need some sort of tool or key for this. Slowly open the valve. Along with the air, hot water will spit out, so be careful. When there's only water coming out you're done.

If it is a steam system, there is an automatic air vent valve on one side or the other about half way up, it looks vaguely like a small bell. It's clogged up for some reason, preventing air from being vented, meaning steam cannot enter the radiator. It either needs to be unclogged or replaced.