My hot water in the bathroom sink is just a trickle. About pencil sized. It isn't the aerator/filter thingy. I clean that regularly, and my cold water is just fine. The water everywhere in the house is just fine. I have two bathrooms. The style of my sink is similar to this one...
Shop Moen Brantford Brushed Nickel 2-Handle WaterSense Bathroom Sink Faucet (Drain Included) at Lowes.com (http://www.lowes.com/pd_89236-866-6610BN_0__?productId=1266019&Ntt=bathroom+faucets)

I have no idea how to fix this.

I turned the hot water valve off, then back on again, and it didn't do anything. There are no leaks under the sink. The tubing for the hot water valve, does look old though. It's some kind of plastic type material. My cold water one is metal material.

It sounds like you have debris in hot port of the valve, close the angle valves under the sink, remove the handle the dome and the cartridge, now put a towel on top of the valve slowly open the angle valve to flush out the debris out, close the angle valve and install the cartridge and see if the problem is fixed,if not then you need to check the hot supply line for and blockage, Thanks.

John