Posture and turnout are the foundations of ballet movements. A neutral spine and correct turnout will allow for a relaxed, fluid upper body, with graceful neck movements and port de bras. Use your inner thigh muscles to support a deep rotation of your hips and draw inward and upward with your lower abs to maintain a strong turnout. The more you practice this posture, the more it will become second nature for your core muscles.
Strengthening the transverse abdominus, the deepest core muscle, helps protect your spine while you are dancing. While the rectus abdominus gives you that "six-pack abs" look, the transverse abdominus is much more important in ballet. A weak transverse abdominus puts unnecessary strain on your lower back. You may have been told in beginning classes to "keep your ribs in" -- this is one way of saying to keep your transverse abdominus tight. Think of wearing a very tight belt or corset pulling your abs in and up.
Core strength is essential to master pirouettes. Without a strong core, your shoulders will twist separately from your torso and you will fall out of alignment. Once you have a strong core, you will be able to turn without relying on momentum. Practice your passe releve to work on the core muscles you need for pirouettes -- standing on the ball of one foot (or en pointe if wearing pointe shoes), hold the other foot at knee level with the hip turned out. This will require you to engage all the muscles of your spine and pelvis.
Many of the movements in a traditional ballet class work the core. For example, battements work the thigh muscles as well as the deep abdominals if done correctly. Imagine a string connecting the top of your head to the ceiling, and lift your leg to hip height without bending at the spine. Moving from plie to eleve in first position will work your inner thighs and your abdominals. Plank poses are also effective for building endurance in the transverse abdominus.