The core is mainly your lower back and abdominal muscles, including the transverse abdominis and obliques (side muscles). Core stability training aims to improve the function of the muscles surrounding your torso so that they give you more protection and support the joints of your lower back. Core stability will NOT decrease abdominal fat, so even if your abdominal muscles are toned from abdominal exercises, you won't see them if there is excess in that area. So in order to lessen you need to focus on your aerobic exercise and nutrition.
All core muscles are important for stability!
Think of drawing your lower abdomen in using the deep muscles of the abdominal wall and hold for several seconds before releasing. Keep your lumbar spine in a neutral position and breathe normally. You can do this standing, sitting on a stability ball or kneeling on all fours. Exercises including the superman, bridge and plank will work many core stabiliser muscles. Many resistance exercises carried out with correct alignment are recruiting your core stabiliser muscles. For example dead lifts and squats, lunges, shoulder presses and cable machines. So while abdominal crunches are a good core exercise, recruiting some of your core muscles (rectus abdominus and obliques), leaving all other core muscles untouched will just cause imbalance. Think of them as a small part of a balanced core workout.