(covers information from several alternate timelines)
The autonomous self-sustaining mobile holo-emitter, or mobile emitter, was a piece of 29th century technology designed to remotely power and enable a single holographic instance away from permanent holoemitters. Constructed of a poly-deutonic alloy unknown to 24th century science, the mobile emitter was approximately the size of a Human palm. (VOY: "Future's End, Part II", "Message in a Bottle", "Drone", "Renaissance Man")
Owned by Henry Starling in 1996, the emitter was either aboard the appropriated Aeon timeship, or derived from the 29th-century technology therein. Starling used the emitter to grant mobility to his captive, The Doctor, while the latter was displaced in time after he hacked into Voyager's computers and stole the hologram, as equipping The Doctor with the emitter gave Starling the ability to use him as a hostage by threatening to destroy the emitter as it held The Doctor's program. After being returned to the 24th century by Captain Braxton, the USS Voyager was allowed to retain the anachronistic mobile emitter, granting The Doctor a new-found mobility and utility beyond Voyager's sickbay and holodecks, able to slip the emitter on and off relatively easily. (VOY: "Future's End, Part II")
The mobile emitter was a wholly self-contained holographic projector, generating the photons and force fields necessary to allow a hologram to physically interact with its environment. The emitter was either affixed to the exterior of the active holoprogram and thereby visible, or it could be covered or enveloped by the program, effectively hiding it from view. (VOY: "Renaissance Man")
The exact power source of the emitter was never revealed, but it would appear to be capable of operating in most conditions without charging; during the alternate timeline of the Year of Hell, The Doctor was forced to rely on the emitter for the better part of a year after sickbay was destroyed in the first wave of the Krenim attacks, showing no sign that this was any kind of serious handicap to him and none of the crew expressing concern that the emitter's continued power supply might run down even as Voyager itself was left in increasingly battered condition by the Krenim attacks. (VOY: "Year of Hell", "Year of Hell, Part II") In another alternate timeline, the entire ship was frozen on an ice planet for fifteen years after a crash-landing killed all on board, but once Chakotay and Harry Kim found the ship, they were able to reactivate The Doctor's program in sickbay and transfer him to the mobile emitter with no sign that the program or the emitter had suffered from over a decade on the ice planet (VOY: "Timeless"). When The Doctor went on a solo away mission to a planet where time was moving at a rate of a day a second, he was essentially trapped on the planet for three years with only himself to rely on and the emitter apparently maintained his program perfectly well during that time, although this may have been due to the temporal anomalies of the planet meaning that the emitter was only operating for eighteen minutes from an outside perspective, or because The Doctor found a way to discreetly recharge it on the planet (VOY: "Blink of an Eye").
Although a durable piece of technology, the emitter could be disrupted by certain conditions; when Voyager spent a month passing through a nebula, prolonged exposure to the nebula's environment affected the emitter's ability to operate, and The Doctor's attempt to circumvent this issue by tying the emitter into the EPS manifold was only a temporary solution. (VOY: "One")
Eminently compatible with 24th century Federation technology, stored holographic programs (active and inactive) could be transferred to and from the emitter with ease, as simply as a voice command. (VOY: "Renaissance Man")
Although B'Elanna Torres had some trouble understanding how the emitter worked, she was once able to reconfigure The Doctor's perceptions via the emitter in order to better scan their environment when Voyager was stolen by the Nyrians and the crew trapped in their artificial biosphere. (VOY: "Displaced")
The mobile emitter's internal power source could also interact with a variety of other technologies to function as an impromptu battery. The Doctor's emitter was kept offline in 2375 while he was stranded inside a subspace sinkhole in the event it would be needed for emergency power. (VOY: "Gravity")
Fifteen years later, after the emitter was rescued from the derelict Voyager in an alternate 2390, Harry Kim used it to power a stolen Borg temporal transmitter after the transmitter's own power source failed, contacting Seven of Nine and preventing Voyager's crash, erasing this timeline. (VOY: "Timeless")
Several Federation-designed holoprograms interfaced with The Doctor's mobile emitter without difficulty, including the EMH from the USS Equinox (VOY: "Equinox", "Equinox, Part II"), a holoprogram of Leonardo da Vinci (VOY: "Concerning Flight"), Reginald Barclay's hologram of himself (VOY: "Inside Man"), Michael Sullivan (VOY: "Spirit Folk"), and the Hirogen hologram Iden. (VOY: "Flesh and Blood")
The hologram Marayna transferred herself into the sickbay's holo-emitters so that she could steal The Doctor's emitter in order to be able to leave the holodeck and roam around USS Voyager, despite the fact that she was being remotely operated by an outside controller. (VOY: "Alter Ego")
In 2375, a transporter accident resulted in The Doctor's mobile emitter being contaminated with nanoprobes. The probes assimilated the emitter, and after sampling Ensign Mulchaey's DNA, it built a maturation chamber and ultimately became embedded in the cerebral cortex of a highly-advanced Borg drone, designated One. After One's death, the mobile emitter was retrieved and returned to The Doctor for his use. (VOY: "Drone")
The Doctor's holonovel Photons Be Free had a fifty-kilogram backpack version of the mobile emitter – intended to reflect the burden of responsibilities that he faced – that the protagonist had to wear to leave sickbay on the USS Vortex. This version of the mobile emitter was one of the prominent features of "Chapter 5 – 'Out of the Frying Pan' (in which our protagonist must confront abusive colleagues)". (VOY: "Author, Author")
In the script of "Future's End, Part II" (both the first draft and the final draft), the mobile emitter was described as "a thin, super-high tech band" which would be worn "across" the wearer's upper arm.
In the Department of Temporal Investigations novel Watching the Clock, the mobile emitter's existence is one of the issues Janeway is questioned on after returning to the Alpha Quadrant, with DTI agents Dulmur and Lucsly criticizing her violation of the Temporal Prime Directive by using future technology so blatantly, only for Janeway to defend her decision by arguing that she wasn't going to deliberately handicap her chief medical officer for the sake of protocol.
In The Light Fantastic, Data and Commander Geordi La Forge talk with The Doctor and Lieutenant Reginald Barclay about the possibility of replicating the emitter for general use, but The Doctor confirms that the emitter cannot be replicated by current technological standards.
In Takedown, Admiral William T. Riker and La Forge are able to create a new mobile emitter while Riker has been augmented by the Cytherians, allowing Riker to accompany Picard to confront the Cytherians by projecting an aspect of his consciousness into the emitter despite his body being 'confined' to the Cytherian chair, but Riker gives the emitter to the Cytherians after the crisis is over as he feels using it would be a roundabout violation of the Prime Directive, as he could only assemble it using resources provided by the Cytherians.
Vic Fontaine infers the existence of the mobile holographic emitter and Morn has one made for him. This attracts the attention of Federation Security. The mobile emitter is used to allow Vic to escape mobsters in his program in Star Trek: Deep Space Nine The Long Mirage, but Vic decides to return to his own program and not to continue using the emitter.
In the timeline of Star Trek Online, The Doctor filed a lawsuit against Starfleet to retain control of his mobile emitter, leading to a movement lobbying to establish civil rights for sentient artificial lifeforms, including holographic (photonic) projections and androids. The Soong Foundation develops a mobile emitter for civilian use, and photonic lifeforms equipped with mobile emitters serve on away teams alongside biological beings. Player characters can obtain photonic bridge officers with mobile emitters through a variety of means, including with Lobi crystals (obtained by opening "lockboxes" with keys bought from the game's store) and through veteran rewards. In the mission "Beneath the Skin", Paul Stamets is recreated as a hologram using Starfleet historical archives, and equipped with a mobile emitter to allow him to accompany the player character into the mycelial network.