Core Stability refers to the integrity of your torso, neck and head. Mainly revolving around protecting the spine and internal organs, the muscles of the core act to maintain optimal alignment of the vertebral column and proper posture. A strong core can also help to provide balance within the body.
The main musculature of the core is considered to be your rectus abdominus (forward flexion and upright posture), external and internal obliques (twisting and anti-rotation), transversus abdominus (upright posture, holds in organs) and the erector spinae group (back extension and upright posture), as well as the pelvic floor muscles (digestion and pelvic stability).
Balance can viewed in many ways. The most obvious example of balance is the ability to stand still on one foot. But balance also refers to the equality of strength & flexibility within a muscle group (for example all four heads of the quadriceps are balanced in terms strength and range of motion)
Balance can also refer to opposing muscle groups such as the hamstrings on the back of the thigh and the quadriceps on the front. Balance between upper body strength and lower body strength is also important, as is balance between the right side and left side of the body or limbs.