The most common stroke a lot of people who are swimming for fitness use is the freestyle, which is sometimes also called the front crawl. This stroke is popular because it's relatively fast, yet the level of effort it requires is sustainable -- so it's useful for building endurance. Other strokes, like the breaststroke or butterfly, are either slower or take too much effort for the average swimmer to maintain for a long workout.
There are other benefits to freestyle as well. It uses the full range of motion for both your arms and your legs. It also engages your core as you work to keep your body level in the water while switching arms from side to side. The freestyle is also a relatively simple stroke to learn -- once you've had a few tips.
In freestyle, most of your body will be just below the surface of the water. You'll want to position your head so that you're looking straight down. The water line should cut across your forehead.
To begin the stroke, start doing a scissor kick, with one leg going up and the other coming down. The motion should start from your hips and end with a little flutter from your lower leg. Don't lock your knees or ankles -- the motion should be fluid.
As you begin the kick, begin the arm movement as well. Reach forward with one arm. Try to avoid bringing it straight up out of the water. Instead, bring it forward with your elbow bent. This is called the recovery phase of the stroke. When your arm is extended in front of your head, slip (don't slap) your hand into the water and cup it. Pull back to your hip, (this is called the pull phase of the stroke) then start to bring the hand forward again. When one hand is entering the water in front of your head, the other should be exiting the water at your hip.
As your arm comes forward, turn your head to that side, look slightly behind you, and breathe. You shouldn't have to lift your head out of the water; turning your head will create a channel in the surface of the water around your mouth, allowing you to get some air. You can make your breathing time more efficient by exhaling while your face is in the water. Also, you don't have to take a breath with each stroke. Find a breathing pattern that works for you.
You want to keep all of your motions fluid and smooth. While some splashing is inevitable, keeping it to a minimum means that your stroke isn't to violent and you're keeping things smooth. Remember to keep your core engaged. As each arm goes forward, your torso should rotate slightly, raising that side. Your head, however, should remain still and shouldn't rotate, unless you're taking a breath.