Prepare the hole in which the basin will sit. If you are replacing a sink and the hole is already there, make sure to clean the area around the hole. Scrape off any leftover caulk, and make sure the surface is clean and dry. If it is a new countertop, cut the hole. Most new sinks come with a template that can be used to mark the area you need to cut out. If there is no template, lay the sink upside down on your counter. Trace a line around the outside. Draw another line inside your first that corresponds to the amount of space you need for the lip of the sink. This inside line is your cut line.
Drill holes in all four corners on the inside of the cut line. The holes should be big enough that the blade for the jigsaw will fit inside. With your jigsaw, make a cut along the top and bottom of the hole. To keep the counter from cracking loose and falling as you cut out the rest of hole, screw a scrap piece of wood across the bottom of the counter, making sure it sticks over the cut by a couple of inches. Cut out the rest of the counter, making sure you cut on the inside of the line. Make sure you get the corners correct, without going outside the line. Once you are finished, you can pull out the countertop and remove the scrap wood.
Dry-fit the faucet and drains into the hole to make sure everything fits the way it should. When you are sure your sink is going to fit and the lip comes into contact with the counter all the way around, you can move to the next step.
Attach the faucet and any other accessories you are adding. It is much easier to attach your faucet and drains when you can see what you are doing as opposed to climbing under the sink to attach everything. Your faucet and attachments should come with instructions, which you should follow.
Place your sink, upside down, over the hole. Line the lip of the sink with your silicone caulk or plumber's putty, making sure the coverage is even all the way around the lip. Flip the sink over and seat it in the hole, pushing down gently. Crawl under the sink and attach any mounting brackets. Go ahead and snug them up. You will probably find that there is excess caulk that has been squeezed out from under the lip. Remove this with a clean, damp cloth.
Attach your drain and water supply lines. Turn on the supply lines and remove the aerator from the end of the faucet. Turn the water on, allow it to run for a minute or so, turn it off and then reattach your aerator. Make sure nothing is leaking and that you have a good seal for the drain. And that's it; you now have a beautiful sink and faucet, ready for use.