Trimmable horizontal stabilizers (THS) are found in majority of airliners and large transport aircraft (like C-17 globemaster III, for example). They are usually part of the trimming system of the aircraft, unlike the elevators, which are controlled by conventional pilot inputs (like yoke etc.). The THS offers some important advantages like:
The required elevator deflection angles are smaller in case of trimmed aircraft and the system has full elevator deflection angles at extreme trim angles.
If the stabilizer is not trimmed, the (human or auto)pilot has to continuously adjust the controls to prevent the aircraft from pitching up or down more than required.
Aligning the elevator with the stabilizer reduces drag.
It allows for a wider range of c.g. movement compared to the elevator-trim tab system.
The decision to use a THS (or not) depends on the design. For example, the BAe 146 had a fixed tail plane in order to reduce complexity (achieved in part by the elimination of leading edge slats). The THS does add more complexity to the system and has been involved in a few accidents, like the Alaska Airlines Flight 261 and China Airlines Flight 140.
The stabilator or all moving tailplane is entirely different (with no elevators) and is mostly used in supersonic aircraft. It is used mainly to overcome the problem where elevator becomes unusable due to shockwaves produced by tailplane and the problem of Mach tuck. Combat aircraft to use them as they create a large pitching moment for lesser control effort. Also, they are used for roll control via differential movement.
In civil aircraft, Lockheed L-1011 Tristar used stabilators.
Image from tristar500.net
It is also used in some GA aircraft (like Piper Cherokee); the strong control response of the stabillator can result in overcorrection- this is overcome in part by using an anti-servo tab.