Vertical launch system
Vertical launch systems (VLS) have become the standard for guided missile launchers in modern submarines and surface warships. They are a distinct departure from earlier launchers that, much like a naval gun, would point the missile in the general direction of the target. Originally motivated by engineering necessity for submarine-launched ballistic missiles (SLBM), developments in missile technology have made VLS much more attractive than earlier, even reloadable, launchers. They are now the only launchers, other than extremely short range final defenses, among modern vessels such as the U.K. Type 45-class destroyers, U.S. Burke-class destroyers and Ticonderoga-class cruisers, and French and Italian Horizon-class frigates. 
The first submarine-launched missiles required the submarine to surface, and used hangars and launching ramps.  These were very inefficient in the use of limited space. UGM-27 Polaris ballistic missiles launched vertically and could be packed fairly densely into the submarine. Vertical launch was more intuitive for a long-range ballistic missile than for a tactical surface-to-air missile (SAM) or cruise missile, since land-based ballistic missiles needed to start vertically, and then gradually adjust their flight paths.
Early U.S. naval SAMs including the RIM-8 Talos, RIM-2 Terrier, and RIM-24 Tartar were replaced by the many generations of Standard missile, starting with the RIM-66 Standard SM-1. All of these missiles launched from "single arm" or "dual arm" rails on mounts that elevated and rotated. The mounts were manually loaded at first, and then went to automatic loading. A fundamental problem was that the most practical means of automatic loading was "revolver-type"; the swiveling launcher, in a visually impressive maneuver, would "bow" toward a storage tube in the deck and be loaded, then snap back to a high vertical angle. Its limitation was that the loading tubes had to be in a circle outside the maximum rotation of the launch, and this was space-inefficient as well as mechanically complex. Plausible deck mountings simply could not hold enough missiles.
Vertical launch seemed counterintuitive for SAMs that were the replacement for anti-aircraft artillery, which clearly had to point at the target. The success of vertical SLBMs did demonstrate a way to have mechanically simple and high density storage.
Other technologies made VLS SAMs practical: improved radars and faster rocket motors, as well as the danger of long-range weapons on enemy aircraft, were pushing the intercept point farther and farther away from the ship. There was time for a vertically launched missile, with a high-energy engine, to alter its course in the appropriate direction without having to be pointed by the launcher.
There are two basic kinds of VLS: hot-launch, where the rocket motor fires in the launch tube and the main rocket moves it out of the tube, and cold-launch, where the missile is pushed out of the tube either directly or by a piston powered by a gas generator, and its rocket motor ignites when the missile is in the air. Each type has arguments and tradeoffs; there are some intermediate variants. 
Hot launch has the obvious problem of causing intensely hot exhaust to be present inside the ship. The heat of launch cannot damage nearby equipment or damage the ship, so there has to be considerable insulation, and mechanisms to vent the exhaust. Hot launch, however, is mechanically simpler than cold launch, which needs a second complex system to pop the missile into the air. Western designers have preferred hot launch while the Russians used cold launch.
Cold launch obviously is safer; Russian VLS actually are angled slightly off the vertical, so if the rocket motor does not fire, the missile drops safely into the sea.
Some of the Russian systems (i.e., for the DIA/NATO-designated SA-N-6 GRUMBLE) use a vertical "revolver" of 8 launch tubes that rotate in front of the cold launch piston. The Chinese implementation, for the same missile, puts a separate cap on each of the launch tubes and eliminates the revolver mechanism. With the exception of the RIM-162 ESSM, which puts a "four pack" into a single Mk. 41 launch tube, the U.S. system is like the Chinese one in that each tube has a separate cap, but unlike in that it is hot launch.
Standard or specialized?
U.S. designers have emphasized standardization on one basic VLS, the Mark 41. A wide variety of missiles have been adapted to it, including SAMs of various ranges, cruise missiles, anti-ballistic missiles, surface-to-underwater missiles, etc.
Britain's GWS 26 VLS Sea Wolf SAM aboard its Type 23 frigates can only fire the Sea Wolf, due to the way the hot launch gas is distributed, very compactly, around the body of a Sea Wolf. The Sea Wolf system, incidentally, has VLS and non-VLS versions.