These poles are generally sized to the arm pit of the skier. They are used for locomotion while skiing, so they should be quite stiff. If you plan on skiing in deep soft snow, you should purchase poles with a large basket.
These poles are generally sized to about the elbow of the skier. They are used mostly for balance (going uphill and downhill) and not as often for locomotion. Most styles will work, and $2 poles from the local thrift shop are not an unreasonable possibility. Beware that the basket on many alpine poles are far too small, and that larger baskets are available.
A useful hint: If you have a set of poles that will be only for ski touring, get them about 3 inches longer than the standard alpine length. Then at the standard alpine length (i.e. elbow height), wrap each pole with enough duct tape to create a comfortable handle. This gives you " adjustable pole" on the cheap. You have short poles for climbing (and possibly the descent) and long poles to help with locomotion on the approach. You will also have a ready supply of duct tape.
Alpine probe poles are adjustable length alpine-style poles. They can be assembled together to form a probe for use in recovering an avalanche victim (among other uses). They can also be adjusted to a short length for downhill skiing and long lengths for climbing or pushing on flat terrain. They are a nice "extra" piece of equipment but not essential.
Generally, avalanche probe poles are not very good avalanche probes, and you might want to invest in an actual avalanche probe (see safety equipment) if you ski in avalanche terrain frequently.
Older avalanche probe poles that used a twist mechanism to adjust the pole length were difficult to use because either they would collapse or get stuck. Therefore, beware of used poles.