The bits of machinery that connect your boots to your skis. Bindings perform two important and mutually exclusive functions. The first is to keep you connected to your skis for as long as possible. The second is to disconnect you from the skis when the forces developed in a fall are likely to hurt you. The basic way they achieve this is spring loading, so that they release when the force of a fall exceeds the force of the spring.
One tip is that, if you eject from your bindings in deep snow so that the ski is under the surface and not immediately visible, your ski will almost certainly be just above or near the top of the impact crater. Use a pole or ski to chop downwards into the snow. Eventually you will hit the missing ski, which can then be dug out.
Links to Binding Manufacturers' Web Sites Although there are many brands there are only a small number of manufacturers as there has been significant consolidation in the industry over the last couple of decades. This is true for most of the ski industry.
Different skiers have different heights, weights and abilities. The force of the spring is adjusted to compensate for this. The likely forces are calculated and set out in a table, and the bindings set accordingly. Unless you are reasonably knowledgeable this should be done by a trained technician.
There is a standard called the DIN setting which relates to the forces needed to make bindings release. Bindings can be adjusted across a range of DIN settings for each individual. They are numbered from 1 to 20, and beyond. The higher the number the greater force is needed to make the binding release. No binding covers the full range, so manufacturers make different models with different ranges of release forces, or DIN numbers. A binding designed for an adult expert may have a range from 8 to 20+, whereas a binding for a child beginner may have a range from 1 to 8. While it may be tempting to set the bindings on a low DIN setting to protect your ligaments a binding set too low will release at inconvenient times and will be a real pain in the bum at the least and may be dangerous at worst. Setting the bindings too high will stop them coming off, but the release mechanism created by your bones and knee ligaments may release early to protect the binding.
Ski bindings also incorporate brakes. These are retractable arms that tuck away under the boot when the boot is in the binding. If there is not a boot in the binding the arms project below the running surface of the ski and (generally) prevent sliding. Brakes are not 100% effective, and skis should still be placed across the fall line when not on your feet.
If you are concerned about whether a particular brand of alpine boot is compatible with a particular binding, relax. The boot/binding interface is stanndardised and all boots will be compatible with all bindings.
Cross Country Bindings
Tele binding have only lateral stability, they have no longitudinal resistance to movement. They retain the boots by forward pressure imposed by cables or rods running under or around the boots and pushing the heels forward. This is derived from the cross-country 3 pin rat trap binding. In fact modern boots still have the holes to attach to a rat trap binding. Most telebindings have no release mechanism
 A very solid binding made from large chunks of aluminium, or sometimes titanium.
A fairly solid binding made from pressed steel. it has a flat plate going under the foot and is hinged at the front. The hinge is possible too far forward, and I managed to twist one of these 30 degrees in a crash.
cables go around the boots and have a clip at the heel to hold them in place, there are spring cartridges to hold them in place.