The Core

You hear about "the core" all the time in fitness. We all know having a strong core is a good thing, but do we really know why, or even what the core is?
The core is commonly thought to be comprised of abdominal muscles, which is true, but there's much more to it than that. Your abdominals are more than just a single muscle.  The rectus abdominis is the muscle sheath running from your sternum all the way down to your pubic bone.  It acts to flex the trunk and is strongly activated during exercises like sit ups and leg lifts. A few sit ups however, is not enough to fully work your core.  You also have internal and external obliques, which are located on the side and front of the abdomen. These are two seperate muscles, with the external obliques laying on top of the internal.  These muscles are activated with twisting and flexing of the trunk, for example, side crunches.
The deepest of the abdominal muscles is the transverse abdominis and it wraps horizontally around your waist.  This is the muscle that's sore after a Pilates class. The transverse abdominis is working when you do an exercise like a sit up, but, since the rectus abdominis is doing the bulk of the work, it won't usually be sore as a result. It takes more precise and specific movements (like those in Pilates) to primarily target this muscle.
Another very important core muscle is often overlooked; the erector spinae. Located along your vertebrae the entire length of your back, they function to extend the spine as well as flex it laterally (standing side bends).  
Often times what happens is people do their sit ups, maybe some side crunches, but they leave the gym without working their back.  This creates a muscle imbalance as the front core muscles are strong, but the back is weak.  All the muscles of the core work together, so if some muscles are not strong, it adversely effects your core as a whole.  
Your core is important for a variety of reasons. First and foremost you use your core in everything you do, they are your stabilizing muscles.  Almost every motion begins at the core.  Core muscles stabilize and balance you in everything from walking to exercising different muscle groups.  Your core also plays a role in posture.  A strong core can result in better posture and lower rates of low back pain. 
Knowing the individual components of the core is beneficial to your workout program.  Make sure when doing core work to include exercises that target all these muscle groups.