Two of the key elements of Pilates are core muscle strength and spinal alignment. The core musculature is loosely defined as the spine, abdomen, pelvis, hips, and the muscles that support these structures. Some of the main core muscles are the erector spinae (located in your back along your spine), the internal and external obliques (the sides of your abdomen), the transverse abdominis (located deep in your gut, this muscle pulls your belly button in toward your spine), the rectus abdominis (the "six-pack"), and hip flexors (in your pelvis and upper leg).
During a Pilates session your instructor will continuously prompt you to concentrate deeply on your core muscles, as well as on your breath, the contraction of your muscles, and the quality (not quantity) of your movements. These are also key elements of Pilates, and your instructor will emphasize them at every session.