If you are replacing a faucet, and have already purchased the new one, look at the manual/schematics and parts included with the new faucet. This is a great way to help you remove the old faucet correctly.
Turn the water supply off before beginning any work! Normally there's valves underneath the sink (one for hot water, one for cold water). Hot is always on the left. You should also be able to tell by the manufacturers label of each side (hot should have a red marking, cold should have a blue marking), or it may just say whether its hot or cold.
After the water supply is off, use a crescent wrench to remove the nuts where the hoses connect to the water supply. Follow the hoses up to where the actual faucet is, and remove those nuts. Then, you can pull the hoses out.
The stopper for the drain is also hooked up to your faucet, so that will need to be removed. Look under the sink. There's a variety of ways manufacturers configure the stopper mechanism; but once you examine it you should have a good idea of how to separate the lever to the faucet from the stopper.
Next, if you look behind the bowl and up toward the faucet under the sink, you should see 2 nuts that have a plastic surrounding, allowing them to be easily removed by hand. Remove these nuts. If its an older faucet, or the original installer tightened them up pretty good, you may need to use pliers.
Now you should be able to remove the faucet easily, and you're now ready to install the new faucet.
Pull the new faucet out of its packaging, and you'll probably notice a small plate provided that the faucet will sit on. Put this plate on first.
Note: For extra protection from water leaks from the faucet into the cabinet underneath the sink, use plumbers putty. The putty, with a consistency of a clay or a hard dough, never gets really hard. Form the putty into a snake and guide it underneath the plate. Make sure to apply evenly, and take care to avoid any gaps. Then place the plate on the sink. Don't worry if some of the putty presses out; take a butter knife and remove the excess after the faucet is fully affixed on top of the plate. You don't need to throw the excess putty away...put it back in the can and reuse it for another job!
Then, proceed in installing the faucet in a similar fashion to the way you removed the old faucet.
Connect your stopper to the faucet. You will want to use plumbers putty to secure any gaps between the stopper plate and the sink itself, in a similar fashion to the faucet plate.
For the supply line hoses, if the length is different from the old faucet to the new faucet, you may need to purchase new hoses. There are a few options. Copper hoses will probably need to be replaced, as they do not provide much flexibility unless you're replacing your old faucet with a very similar new faucet.
There are hoses that are braided and come in different lengths. They're available at your local hardware store are probably the easiest to put on, although it might cost you a little more. Make sure you measure the distance between your supply line and your faucet to ensure you get the correct length of hose.
Once you attach the hoses, the next step is to ensure there are no leaks. This is done by trial and error. First, tighten a little more than hand-tight, and turn the water supply on. If there are no visible leaks, run your finger around the nut to make sure its dry. Do not over-tighten!
Adjust the stopper to ensure it holds water when the lever by the faucet is pulled up all the way, and you should be all set!