Most gyms have some type of back extension equipment, be it a Roman chair or a larger machine with attached weight stack. Whether this is a good or bad exercise for you depends on how you do the exercise and the status of your back. If you have current back pain or a recent injury, skip the back extension machine until you're healed.
Back extensions use the erector spinae, a three-part muscle comprised of the iliocostalis, spinalis and longissimus. The erector spinae is responsible for both flexion and lateral flexion of the spine. Done properly, back extensions specifically target this muscle, strengthening your lower back.
If you're using a back extension machine with a weight stack for the first time, start with a low weight and work on your form. Sit on the seat and position the roller against your shoulder blades. Slide your shoulders down and keep your chest lifted as you lean back. Don't go all the way to horizontal. Return to your original position with control. Over time, you can slowly add weight. In his book "Strength Training Anatomy," Frederic Delavier calls back extensions a great exercise for beginners. He recommends doing sets of 10 to 12 repetitions. Monitor how your back is responding. If you feel pain, this might not be the back exercise for you.
Back extension machines are controversial among some fitness professionals. Personal trainer Mike Behnken put them on his list of worst weight machines. He equates using a back extension machine to lifting something heavy without bending your knees. Some trainers oppose exercises like the back extension machine that isolate one muscle or muscle group, preferring those that necessitate several muscles working together. Behnken wrote on his Ask The Trainer website that people like back extension machines because they mistakenly believe they can easily burn off love handles without the extra effort required for the exercises he considers more effective.
Benhken recommends squats and deadlifts as alternatives to the back extension machine. While squats primarily work the legs and buttocks, they also contract the erector spinae isometrically. Or you could do a simpler back extension on the floor. Lie prone on your stomach with your arms in a V behind you, like wings. Use your back muscles to lift your chest, repeating the movement 10 to 15 times. If your gym has a Roman chair, this is another alternative. With this apparatus, you'll lie facing down with your hips resting on the large pad and the area just above your heel hooked under the foot pad. You'll start with your torso hanging down and then lift your upper body, using your back muscles, until it's parallel to the floor, being careful not to overextend your spine.