As a balance training aid, the bosu is pretty easy and thus a nice place to get started (progression in roughly increasing difficulty: standing on Bosu, kneeling on a fitball, standing on bongo board, standing on styrofoam roller, riding unicycle...)
I'm a bit fanatical, since I'm 48 and race and live at sea level (originally typed "fat sea level" and, heck, that's an issue too), but some nice exercises on a bosu (except for the last one, all flat side up) are (if you can do them safely--you must calibrate for your fitness level and current balance):
Juggling (balance, recovery; progress until you can do this kneeling on a fitball and then standing on a styrofoam roller; instead of juggling tennis balls you can toss a medicine ball from hand to hand.)
lawnmower pulls with a hand weight (hold weigh in right hand to the left of your left ankle, knees bent, then straighten and cross your body and raise the weight so you finish with it held above you and to the right with the right arm fully extended. "Stayin' alive...")
Hold a tuck while holding a medicine ball in your hands. Work your way up to two or three minutes.
High tuck-low tuck-standup-scream. (At the end of a leg workout, just to make sure your legs are good and thrashed, holding a heavy medicine ball, go from standing to a high tuck, to a low tuck, to a high tuck and standing again, and keep repeating without a break; to make it more of a balance challenge, toss the medicine ball from hand to hand while you do this.)
Haybalers: Take a medicine ball in both hands, and with your knees deeply bent, hold it to the left of your left ankle. Then stand up, pulling it up and across your body until you are holding it above your head and to the right. Repeat 10-16 times on each side. If your knees aren't good, this may not be safe for you, but it really does create some of the same knee positions and demands as slalom skiing.
Agility/anaerobic threshhold hops: Do this on a surface with some give to it, to save your ankles and knees. With its flat side down, stand to the right of the bosu holding a medicine ball, with your left foot on top of its rounded top and your right foot on the ground. Hop lightly over the bosu, and finish with your left foot on the floor and your right foot on the rounded top of the bosu, and immediately hop in the opposite direction to repeat.
Once the first 5 get too easy on a bosu, you can do them on a bongo board instead. Bosus and bongo boards are also nice to add a leg/core/balance component to most upper body lifts (if you can do so safely), so with the flat side up you can do upright rows, bicep curls, military presses etc. while standing on the bosu.
Balance is a key component of high level skiing, and in my experience balance improves much more with training than does strength or power. But like strength training, to keep getting better you have to introduce more and more difficulty into your balance challenges as you progress.