The connections and attaching hardware can be pretty corroded and hard to break loose.  Tools that can make the job easier are channel lock pliers and a basin wrench.  A basin wrench allows you to reach up behind the sink or lavatory and loosen the nuts holding the faucet in place.  Penetrating oil can also come in handy.

The first thing you will have to do is to turn off the water to the fixture.  Hopefully the shut off valves under the fixture work properly.  If not, you will have to turn of the water supply to the house.  After the water has been turned off, you can disconnect the supply lines from the faucet.  The next step is to loosen the nuts holding the faucet in place. 

After you have removed the mounting hardware, the faucet should be easily removed from the fixture.  You will want to clean up the sink surface before installing your new faucet.

 If the new faucet doesn’t come with a gasket, you will need either a good grade of caulk or plumbers putty (do not use plumbers putty on cultured marble or fiberglass) to put between the bottom of the faucet and the fixture to keep water from seeping underneath.  If the faucet has a gasket, place it on the bottom of the faucet and set the faucet on the fixture through the holes provided.  If using caulk or putty put a small bead around the perimeter of the base of the faucet, then put it in place.  Most two handled faucets are anchored with large nuts screwed onto the shanks that the water connects to.    Many faucets provide nuts that can be easily tightened by hand.  If the nuts are not that type, a basin wrench can help.  A pair of adjustable pliers can also be used.  Some single lever faucets install in the same manner.  Others are mounted with nuts and washers.  Single hole faucets have many different mounting methods, consult the instructions for details on mounting.

The next step is to connect the water supplies to the hot and cold connections.  If the existing supplies are in good condition and the correct length you may be able to use them over.   If not you will need to purchase new supply lines at your local hardware or plumbing supply store.  Make sure that you get the right combination of connections on the supply lines to fit both your shut off valve and faucet.  There are several different sizes on the shut off valves, but a majority of faucets are ½ inch.  If you are using copper tubing to connect the faucet, make sure to use fresh washers on the connections.   Compression (ferrule) connections can usually be resealed.

 After connecting the water supply lines, you should turn the water on at the shut off valves.  It is a good idea to remove the aerator at the end of the spout before you turn on the faucet.  This will allow you to purge the system of debris when you first turn the water on.  After allowing the water to run for a short time turn the faucet off and inspect your connections for leaks.   If you find any, try tightening the nut and recheck.  If this doesn’t fix the problem you may have to redo this connection.

 Clean up any excess caulk or putty and you are done.