The official definition of concentric exercise is a contraction that shortens a muscle, while eccentric exercise is a contraction that lengthens the muscle; that all makes it sound more complicated than it is. Most exercise is concentric focused. It's the part of the exercise when the targeted muscle is working to perform the action. The eccentric portion is the return.
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To make it even simpler, think through some standard exercises. When using a leg extension machine to work the quads in the front of your thighs, the concentric portion of the exercise is the part where you extend your leg against the resistance. Even though your leg is extending, your quad muscles contract to straighten your knee. As you return to the bent-knee position, your quads muscles extend. Similarly, the concentric motion of the biceps curl is when you bend your elbow and bring the weight toward your shoulder. The return to the starting position is eccentric.
Another way to think of concentric exercises is moving a weight away from the pull of gravity. The eccentric phase is controlling the weight against gravity's pull. With a pushup, the concentric phase would be when you extend your arms and push your body away from the floor. With a body-weight squat, it's when you extend your legs to a standing position. In both cases you are working against the pull of gravity as you move your torso away from the floor and controlling its pull to keep you from hitting the floor on the return.
The eccentric phase of an exercise isn't useless. Done properly, in a slow and controlled motion, your muscles work in the eccentric phase to slow the motion down. This both avoids injury and works your muscles in a different way. Eccentric training is sometimes used by elite athletes and lifters. Of course, the only way an exercise can be fully eccentric is if a partner hands you a weight at the top of an extension. Bringing the barbell or dumbbells down to shoulder height would be eccentric, in that case. The most common way to perform eccentrically focused exercise is to use explosive movement, making use of momentum in the concentric phase, then returning slowly.
Yet another form of muscle contraction is isometric, where the muscle remains static, meaning it doesn't extend or contract. An example of this would be a plank. In this case you get into a classic pushup position, or rest on your forearms with your elbows under your shoulders, and contract your abs to hold your body in a plank position -- with head, back and pelvis aligned -- for 30 to 60 seconds. You're still working against gravity, but to keep your body in one position. To build strength, 180DegreeHealth suggests combining all three types of contractions into your workout.