You don't need to join the Marines or the Navy to enjoy the many benefits of doing a proper push up (sometimes referred to as a press up). A basic push up does not require any equipment other than your own body weight and your arms. It can be done anywhere there is a firm surface. And it works the chest, the shoulders, abs, and the triceps. So, it is an excellent exercise for general upper body strength. It can also be varied to meet many needs.

[edit] Steps

Remember to always warm up before any exercise. Warming up reduces the risk of injury and gets muscles ready to do a push up. You can actually lift/push/pull/etc more if you go through a proper warm up routine, as compared to diving straight into the exercises. Make sure to stretch your arms and wrists - key joints in push ups.
Assume a prone position on the floor or other rigid surface (preferable carpeted-more on that later) that's able to support your body weight. Keep your feet together!!!
Make two fists and place them under yourself on the floor, approximately shoulder width apart. If you are on a relatively cushioned surface, such as a carpeted floor, simply support yourself on your fists between the first and second knuckles. If you are on a less forgiving surface, consider investing in some pushup grips (they look like handles you put on the floor.) NEVER do a pushup with your palms flat on the ground, as doing so puts your wrists in a very unnatural position with all your weight on them, which can cause them severe discomfort, strain, and permanent damage. Curl your toes upward (towards your head) so that the balls of your feet touch the ground.
Raise yourself using your arms. At this point, your weight should be supported by your hands and the balls of your feet. Make a straight line from your head to your heels. This position is called "plank," which is used for other various exercises. This is the beginning and the end position of a single push-up.
Lower your torso to the ground until your elbows form a 90 degree angle. Keep your elbows close to your body for more resistance. Keep your head facing forward. Try to have the tip of your nose pointed directly ahead. Draw a breath as you lower yourself.
Raise yourself by attempting to push the ground away from you. Breathe out as you push. The power for that push will inevitably come from your shoulders and chest. The triceps (the muscle on the back side of your upper arm) are also contracted but the primary exercise for the triceps isn't the push-up. Continue the push until your arms are almost in a straight position (but not locked).
Repeat steps 5 and 6 for the remainder of the exercise.
There are three types of push ups that use different muscles. Close-hand, regular, and wide-arm. The closer your hands are together the more you will engage your triceps, the wider apart the more you will engage your chest and shoulders.
Stretch the chest and shoulder muscles during your cool down cycle. Proper stretching and cool down routines are just as important as the warm up, but, unfortunately, are quite often overlooked.

[edit] Tips

Normal push-ups are quite difficult to do with good form and proper control, especially so for someone who is just a beginner. If you find yourself shaking slightly as you do a slow and proper push-up, you are doing push-ups that are too difficult for you (or you haven't warmed up enough!)
To make push-ups easier, balance on your knees instead of your feet (you may need an exercise mat) (see Vary the Push up Routine). Also consider inclined push-ups, where you put your hands on something raised, or even do push-ups against a wall.
Concentrate on engaging your chest muscles, squeezing them at the top of the push-up. This builds muscle much faster. If you can't squeeze your chest muscles, do easier push-ups where you can. Consider doing inclined push-ups in front of a mirror so you can watch your chest muscles and be sure they are engaging. Try to eat a small amount of food first.

[edit] Warnings

As with any strength training exercise, if you feel intense and/or sudden unexpected pain in your chest and/or shoulders, stop immediately! If the pain is in your chest and/or shoulders, you have either done more push-ups than you can handle or you aren't ready for the exercise you are undertaking. If it is the latter, start with lighter exercises that target the chest before attempting the push-up. If the pain is somewhere else, you are doing something wrong. If the pain persists, consult a physician.
Positioning your hands closer together to make the push-up harder has diminishing returns. When you want to make them be really hard, try putting your hands together so that they form a triangle. However, if you put them too close together, you might have trouble balancing your torso during the lift and put extensive (and unnecessary) strain on the bones of the arms and shoulders. This might lead to aching of the bones well after the exercise or problems in the shoulder joint in the long run. The danger zone varies from person to person and from one body type to another. A general guideline to follow is: When you place your hands on the ground, extend your thumbs inward toward the opposite hand. If your thumbs are touching each other, you are at the limit. If you want to place your hands further together, consider the other mentioned methods of making the push-up harder. Trying to clap when you come up with your arms straight is another good variation of the push-up. When you do this, though, make sure you hold your tight, straight, position.
Stop doing your push ups when your lower back gets tired. Don't sag in the middle as this could lead to injury!