The sticks that people carry in their hands. Their primary function is to assist in the timing of turns by rhythmic planting, and as a means of maintaining good body position. Poles have numerous secondary functions, such as whacking errant children, impromptu sword fights, pushing along flat bits, aids to the prevention of other people gaining unreasonable advantage in lift lines, forming crucifixes for the discouragement of vampires, handles for towing small children and stranded snowboarders and any number of other functions.
To work out the right length for you, put the handle on the ground with the tip pointing up. Grip the pole below the basket (between the basket and the handle) with your thumb and index finger touching the basket. The correct length for you is the pole where your forearm is parallel with the ground (or your upper arm and forearm form a right angle - same thing). Some people prefer a slightly shorter pole as it helps them get forward and lower in turns. This is a matter of individual style. The good news is that shortening a pole is the work of moments, so follow the basic rule at first. If you feel you want to get lower have your poles shortened (note - poles cannot be lengthened again, so be sure you want to go with the shortening before you commit.)
Most poles are made of aluminium. There are two grades used - one bends, the other breaks. Some poles are made of resin/fibre composites such as carbon fibre. These are light and thin, but cost a lot more than aluminium. They look groovy. Poles used by downhill racers are curved to fit the racer's body in a tuck to reduce wind drag. Do not be tempted to think that it is cool to ski around with these poles. You will look like a plonker. People will laugh out loud.
There are some pole straps with a release mechanism, so that the pole and you are separated in a fall. One of the most common ski injuries (if not the most common) is a strained thumb caused by the pole straps pulling the thumb during a fall. The release straps may not be a bad idea. It is also a good idea to take you straps off if skiing trees. If the basket hooks up on a stray branch only your pole will be left behind, not your arm.
When you grip a pole put your hand through the strap from underneath. When you grasp the pole grip you should also grasp the strap. This means that the strap runs across the back of your hand. As you push with the pole, the strap tightens, meaning that when you push the strap is holding your hand to the pole. It means that you have a firm hold without the need for gripping too hard.
Another school of thought is that the you should not have the loop of the strap between your hand and the pole grip, rather it loops around under your wrist. This will prevent the strap catching your thumb and bending it back in the event the pole goes wild.