On an electric guitar, a tremolo system, often called a "trem" or in certain situations a "whammy bar", is a device, integrated into the bridge and/or tailpiece of a guitar, that allows the player to slacken or stretch all of the strings of the guitar evenly, thereby lowering and raising the pitch of a chord being played. The initial intent was for this to be used for a subtle vibrato or tremolo effect, hence the name, replacing techniques like pushing forward or backward on the neck, or pulling on the strings. The devices, however, became used to much more extreme effect beginning in the '60s with guitar legend Jimi Hendrix, and continuing into the 80's metal and punk genres with other guitarists like Eddie Van Halen and Tony Iommi copying the extreme bends of Hendrix for dramatic effect.
There are three major styles commonly seen on modern guitars:
Bigsby - The oldest style, this tremolo typically replaces the tailpiece of the Gibson and Gretch designs with the floating-style "Tune-O-Matic" bridge, and is the only major tremolo design compatible with these body styles. The tremolo mechanism is a cylinder around which the string is wrapped partway before being anchored to it, and is controlled by a spring-balanced, flat metal lever extending under the strings. Pressing the bar toward the guitar body slackens the strings, while pulling it away stretches them. The Bigsby is standard equipment on most Gretch Electromatic guitars, who own the trademark on the design, and is a very popular licensed factory option or aftermarket add-on among fans of Gibson-style guitars such as the Les Paul and SG. It's not compatible as such with Fender-style bridges, however hybrid instruments have been custom-built which incorporate a typical Fender body style like a Tele with a Tune-O-Matic bridge and Bigsby trem. The advantages are good tuning stability (the trem has a fixed "center point" which allows the guitar to return to the same proper tension after using the trem) and durability, while the downsides include less capability for really bending pitches in the heavy metal style; it's more a true tremelo than a "pitch bender".
Synchronized - These tremolos were designed as an upgrade for Fender's Stratocaster guitars, which originally had stamped-metal "hardtail" bridge plates borrowed from the Broadcaster/Telecaster that both formed the tailpiece to anchor the strings, and held the bridge saddles and adjustment screws. The Wilkinson-style tremolo maintains this basic bridge design, but the bridge plate is attached to a cast metal base that extends down into a cavity routed all the way through the body of the guitar, where it forms an anchor for a set of springs attached to the body, countering the tension of the strings. The design was originally intended to have the bridge plate lie flat against the body while the trem was at rest, which made the tremolo only able to lower the pitch of the strings, and was also intended to use five springs inside the body to counter the strings' tension, making the trem feel very firm and the effect it produced very subtle. However, players soon found they could remove a spring or two, and/or lower the tension of all springs by backing the anchor cleat out of the body of the guitar, which caused the tremolo to rise off the "deck", softening the trem's feel and allowing it to raise note pitches as well. Fender began setting up their instruments at the factory this way soon after the modification became popular. This flexibility came at the cost of reduced tuning stability and sustain; Stratocasters with this trem setup are harder to get into tune and keep in tune, and the lack of a firm center point also means the changing tension of the string as it vibrates sets up sympathetic vibrations in the entire tremolo system which absorb the energy of the vibrating string.
Floyd Rose - This is the overall newest major style of tremolo, invented by Floyd P Rose in 1976, a jeweler by trade and semi-professional guitarist in the evenings, who encountered the aforemntioned tuning problems with the Wilkinson tremelo on his '57 Strat. He first invented the locking nut, which kept the strings from moving through the nut during pitch bends and therefore kept the instrument in tune. He further improved the overall design by changing the anchoring system of the Fender bridge from the series of screws used on the Wilkinson to a post system somewhat resembling that of Gibson tailpieces; this allows the tremelo more freedom of movement, and also allows the system to more readily return to its balance point. Floyd's design quickly became popular after being endorsed and installed on the guitars of legends like Eddie Van Halen and Steve Vai, who still use the system today. The disadvantages are that this free-floating design is affected drastically by the breaking of a string, and that the increased movement capability of the trem requires more of the body to be milled out for the tremolo system, even more severely than for a Wilkinson. The locking nut can also cause issues when players forget the trem is locked down; attempting to retune the guitar by loosening or tightening the tuning machines will have no effect until the nut is unlocked, and in the extreme can cause broken strings as players crank on the tuning keys. Modern Floyd Rose tremolos have "fine tuners" on the bridge, which allow small variations in string tension without having to unlock the nut. A further refinement to the design is the locking tuner, originally developed by Bob Sperzel, coupled with the "roller nut". Instead of locking the string down at the nut, the tuning machine is instead designed to lock the string at a given tension, and then the nut is redesigned with ball bearings that ensure the string glides through the nut slots smoothly as the tremolo is manipulated.
Other systems include the Gibson Vibrola, a Gibson-developed design which attempted to get around the patent protections of the Bigsby (Gibson would eventually abandon this system and license the use of the Bigsby); Kahler, a hybrid of the Bigsby and Floyd Rose developed for Gibson-style guitars; the Fender Floating Trem, a more integrated style of vibrato mechanism which utilized a separate bridge from the tailpiece assembly; and the Stetsbar, which utilizes a cam-actuated tailpiece that moves back and forward as the tremolo arm is manipulated.