Can you explain the difference between balance and stability, if there is one? When someone has poor balance, what plane should I focus on strengthening to make him/her more functional? What plane should I work a dancer in to make her more functional?
To answer your first question, this depends on the context from which one uses these terms. "Balance" may refer to the body's ability to stabilize statically and/or dynamically. It may also refer to muscle synergy in length tension relationships, or postural "balance." The list goes on. "Stability" may also refer to the body's ability to stabilize statically and/or dynamically. It may also refer to joint integrity, etc. In general exercise terms, the differences may not even be worth mentioning.
Your initial training protocols will depend greatly on your assessment(s)/ evaluation(s). Find the individual's balance threshold, which basically reefers to the most challenging exercises/positions/etc. that the individual can perform in perfect posture and alignment. There are countless single leg progressions on the PTN Exercise Library.
Depending on the type of dance, any and all planes of motion will be important to train and progress through. Once again, before exercises can be chosen, the assessment(s)/evaluation(s) must be performed. To be an effective trainer, one must posses at least basic assessment skills. Some up education may be warranted here, so please read the "related articles" below and at right.