If you are not living in the desert where the cold water is hot during the summer, check to see if perhaps the Plumbing is backwards: is the hot water faucet giving cold water? I once had hot water in my toilets.
Because the valve under the sink isn't working. It could be plugged up and not closing all of the way or could be the actual valve you turn broke from the shut off assembly inside of the valve. You will have to find the other shut off for the hot water. Hopefully it is right at your hot water heater or you may have to shut off the water to the whole house. And while you do that, replace the valve under your sink so you don't have to do it like that again. Usually you can buy one for 10 bucks at the hardware store.
Seperate valve for outside faucet turned off, broken valve stem inside of outside faucet or handle stripped at faucet.
The water valve under the sink is turned off or has gone bad. It is possible that there is a blockage in the faucet itself. I'd start by ensuring that the water valve is turned off, and take the faucet apart and look for anything out of the ordinary.
if it is a single lever faucet the cartridge needs to be replaced. check the supply valve under the sink first,
It should have both. Verify that it has two lines to the faucet. If it does, check to see if the stop valve under the sink has been turned off. It may have been leaking and turned off instead of being repaired.
mixing valve in a faucet: ok I will assume you are refering to a shower valve, that is the only faucet w/a mixing valve: shut off the hot and cold water. cut out valve or unsweat it. clean and flux new valve, pipe and fittings(if used) sweat in new valve
it is a tool that resurfacese the seat on a water faucet valve
Turn the water off to the faucet. Remove the handle. The top of the faucet should unscrew to get to the ball valve. Replace the valve and gaskets then reassemble the faucet.
Outside at the main shut off valve. Showers/Bathtubs do not normally have a shutoff valve like sinks and toilets. The main water valve for the house must be turned off.
Possibly unscrew your aerator at the end of the spout and see if the screen is clogged. Blow it out and rescrew the aerator on. If that isn't the problem, sometimes things get caught in the spout. Unscrew the spout and turn the water on. If there is water coming from there and not out the spout when it is hooked up, the spout is plugged. If no water coming out when the spout is off, rehook spout and change cartridges or stems in the faucet AFTER shutting the water off. Depends on the brand of faucet, check the instructions for the brand and model. Also your kitchen dishwasher and sinks are run on seperate valves. The dishwasher valve is usually under the sinks (causing the water to go to the dishwasher first) the valve that was installed for the sink (before the dishwasher) may be bad too.
The water meter is shut off? The main water line is shut off? The valve for that faucet is turned off? The waterline is made of galvanized piping and has corroded shut? Aerator is clogged, remove and clean or replace.
tap valve
if you look under the sink, you will probably see a cutoff valve going up to the sink faucet. turn this on. if that does not fix it, then change you faucet fixture.