The only connection between ski and boot is the binding. This connection system has reached a superior technological level over the years, which tries to guarantee a maximum of safety in case of a fall. Apart from connecting ski and boot, this is the main goal of a ski binding: to prevent injuries.
When skis were still made of wood, the ski and shoe were mutually connected by iron braces and leather straps. Skiers would have a high risk of getting injured when they would fall, because the skis would not let go. Particulary knees and ankles suffered a lot. Nowadays safety bindings cause the boot to disconnect from the ski when it undergoes a certain push force, such as the sideways push of the boot which indicates too much torsion of the leg.
Bindings consist of two main parts. The tip is the forward part of the binding, in which the boot should be snapped first. Modern tips contain a mechanism to adjust the binding to the weight of the skier. The heel is the back side of the binding, which is often provided with a lever which can release or fix the boot. Most of the time both parts are connected by a metal or synthetic interface, which is usually screwed onto the ski. Very modern bindings without interface do exist (so you have two seperate parts then) - freestyle skiers use air pads instead to absorb the shock from landing airs. Flexible interfaces, that shorten and elongate as the ski flexes, can also be found. Most common skiers on the other hand do not want too flexible interfaces. The binding should transfer the movement of the skier's boot to the ski in order to make 'natural' turns and moves. Excessive elastic distortion is particularly unwanted by experienced skiers.
Bindings in the strict sense of the word pretty much only apply to alpine skiing. Free heel bindings exist for telemark and cross country skiers (who are otherwise fixed to the skis entirely). They need to be able to raise the heel with the ski staying in a horizontal position.
Many main ski brands also have a collection of ski bindings, such as Rossignol, Salomon, Marker, Tyrolia, Fritscha, Silvretta.
Thanks to SharQ for his remarks