Skis come three ways:
Flat- No plate, rails, nothing. Bindings are mounted directly to the ski, drilling directly into the core. Usually found on race skis, junior skis, freeride skis, and twin tips. Race skis that come flat are designed to work with a plate, like a VIST or Marker piston control. Twin tips and freeride skis come this way to preserve snow feel underfoot.
Plate- A plate that is just drilled into the core of the ski. May or may not have free floating properties that allow the ski to maintain a natural flex under the binding. Most require a certain binding. Usually found on race, intermediate, and advanced skis. Race skis do this to increase both torsional and longitudinal stiffness and provide lift. Intermediate and advanced skis do it to provide freefloating properties without the cost of integration.
Integrated- Core is milled and a rail is mounted into it, thus providing the best power transfer and smoothest flex. Found on expert carving skis. Expensive.
As mentioned above, some systems only work with certain bindings. the affiliations are:
Marker: K2, Volkl, Nordica, Blizzard
Look/Rossignol: Dynastar, Rossignol
Tyrolia: Head, Fischer, Elan
VIST: Stockli, 4FRNT
Atomic and Salomon both use their own bindings. Line used their own bindings back when they were still in production.
A quick rundown of each binding brand:
Marker: Poor amount of brand loyalty due to former prerelease problems, now solved. Flagship technology is "piston control." Uses a piston at either the toe or midsole to dampen vibrations from the tip. Marker claims only "true mechanical" toe which causes low elasticity and increased toe release. Pairs with Nordica XBS, Volkl IPT, K2 M1, and whatever the Blizzard system is called.
Salomon: Generally good brand loyalty. The original system binding pared with old Pilot technology. Adjustable wings on "S" series cause problems for some with prerelease, but is easily solved with some loctite/epoxy. New "Z" binding boasts light weight and high rigidity and s used with new Smarttrak system.
Rossi/Look: Highest amount of loyalty due to now discontinued Pivot heel. Still uses turntable heel in race bindings. High elasticity, highest forward pressure. Look claims the Pivot was discontinued so a binding with more forward pressure could be developed; real reason is most likely to create a system-compatible binding. Fairly heavy.
Tyrolia: Growing market share because of success of Mojo 15 for freeriding and FF17 for racing. Diagonal release, zero residual fricton AFD. Mojo is light. FF is pretty heavy.
VIST: Heavy. Previously used by Nordica until dumped for Marker. You probably won't see these much.
Atomic: Heavy binding, poor rep as a result of recalls several years ago concerning springs that caused massive amounts of prerelease. Atomic will now be using Salomon Z bindings.
Elasticity- How far the boot can travel in the binding before being released. Highest in Look, lowest in Marker. High elasticity translates to good freeskiing/bump performance where the binding is continuously being subjected to high forces, low elasticity translates to more reliable, but also more frequent, release.
Rigidity- A binding shouldn't twist or flex laterally. This allows power to be effectively transmitted to the ski. Highest in Salomon and Look, lowest in Tyrolia.
I may have missed a few things, but that's the gist of it.