How to Remove Your Old Faucet
Once you have your tools and parts gathered together, you can remove the old faucet. The first thing to do is to make certain that you shut off the water to the fixture. You can usually do this at a pair of valves beneath the sink - provided they exist. Many older homes do not have shutoff valves installed. In this case or if you are replacing leaky valves, you will want to shut off the water where it comes into the house. If you don't already know its location tracking down the main shutoff vavle may require a bit of detective work.
If possible trace your cold water pipe to the point where it enters the house. Usually, you can do this from the basement or crawlspace. If you don't run into a valve by the time the pipe enters the outside world, the valve is probably outside. If you have one outside, it is normally burried in the ground below the frost-freeze line. In warmer climates it may be accessed and turned with a pair of pliers. However, in colder climates it will normally be a socket near your side-walk or curb, marked with a "W". It can be turned with a socket wrench with good leverage.
When the supply water has been secured, turn on a faucet above and below where you are working (if possible) to drain away water in the system that might come gushing out when pressure is released. Also turn on the faucet you are removing. Now is when your basin wrench is going to come in handy. Flop the head to the side, and fit the teeth around the connection between the hose (or pipe) and the faucet. To determine which way to turn, pretend you are opening a jar upside down, or if this is hard to imagine, turn the nut counter-clockwise. You will probably have to lie on your back under the sink, within a cabinet, so be sure to clear out whatever is under there before starting. Also, you will be working around the basin and the drain. Be sure not to loosen the drain inadvertently, and keep a good grip on the basin wrench so it doesn't drop on your head, which will likely be positioned almost directly below it. (This sounds worse than it really is - just take common sense care and you will separate the faucet from the hoses in a matter of moments.)
Up and beyond the connection with the hot and cold water supply hoses, you will find two washers. These washers secure the faucet to the basin or cabinet. They are usually only on hand tight, but if not, you might have a go at them with your basin wrench. They must be spun completely off the pipe. In a bathroom sink, there is normally a bar hanging down that allows you to open and close the drain plug from a knob on the faucet. Remove this from the drain plug by squeezing the securing clip and sliding the bar off the drain plug lever. Now the faucet should lift right off the basin.