Place your hands properly. On a standard English keyboard, your left index finger will be resting on the F key. Your other left-hand fingers will be resting on D, S and A, respectively. On the right side, your index finger will be on the J key, with the other fingers on K, L, and ;. Each thumb should be on the space bar. Most modern keyboards have small plastic nubs on the F and J keys as a tactile reminder that your fingers are in the correct position.
Learn which fingers press which keys. Reference the chart to do so. (The left index finger extends up to press both R and T; the right index finger both U and Y; the left middle finger E; the right middle finger I, etc.)
Start typing (and watching). At this point, many typing courses would have you complete numerous exercises involving random keystrokes and random paragraphs. While this is helpful in learning finger movements, it's extremely boring. To make the learning process less tedious, simply start integrating touch-typing into your everyday keyboard use:
Type without looking. After you have grown somewhat comfortable with the finger movements, it's time to start "no look" typing. This will seem tricky at first, and you may have to reference your keyboard occasionally to remember the location of certain keys. Even if it slows you down, try not to look.
Practice, practice, practice. Once you have become good enough to do most of your typing by touch, rather than sight, keep practicing. The more time you log at the keyboard, the better a typist you'll become.