This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
The holo-communicator, or holographic comm system, was a type of communications technology employed by the Federation and the Klingon Empire. It allowed two parties to converse as if in the same room by sending holograms of each other courtesy of projectors.
During the 2250s, the USS Shenzhou had an early version of this type of device installed for use on the bridge and in private. It was prone to glitches. (DIS: "The Vulcan Hello", "Battle at the Binary Stars")
In the same era, the technology was also in use by the Klingons. Their version of the device produced monochrome images when used between Klingon ships, but more color when communicating with a Starfleet one. (DIS: "Battle at the Binary Stars", "Point of Light")
After the USS Enterprise suffered a critical systems failure that crippled the starship, the crew determined that the holographic communications system had been the cause of the shipwide failure. Number One reported that the holo-communicator had been responsible for the ship's problems to Captain Christopher Pike. Pike, who disliked the holo-communicator anyway because the holograms "[looked] too much like ghosts", ordered that Enterprise's Chief Engineer Louvier remove the system, stating that they would use viewscreens going forward. (DIS: "An Obol for Charon")
In the 24th century, they were included on the bridges of ships such as the USS Defiant and USS Malinche; Captains Sisko and Sanders used theirs for the first time to coordinate an attempt to capture Maquis leader Michael Eddington. The Maquis had been able to "procure" a holo-communicator of their own, allowing Eddington to use it to contact Sisko during the hunt. As the communicator was installed after a cascade virus that damaged all other systems on the Defiant, it continued to work throughout Sisko's pursuit of Eddington. A holo-communicator was also installed in Captain Sisko's office on Deep Space 9 at this time, and was used by Odo to communicate with Sisko aboard the Defiant. (DS9: "For the Uniform")
The holo-communicator in Captain Sisko's office was later used to allow him to converse with Rear Admiral Bennett on the matter of Doctor Julian Bashir's genetic engineering. (DS9: "Doctor Bashir, I Presume")
Kurros of the Think Tank used a similar technology, which created an isomorphic projection of himself aboard the vessels of potential clients. The projection was sophisticated enough to taste beverages. (VOY: "Think Tank")
Shinzon used a holo-communicator to demand Jean-Luc Picard's surrender in 2379. His device was more advanced, however, as it was able to be projected into an area without holo-emitters and it was impossible to trace the signal back to the Scimitar. (Star Trek Nemesis)
Background information Edit
The idea for the holo-communicator was Ronald D. Moore's. According to Moore, "That's something I had been pushing for because I just think it's so absurd that in the twenty-fourth century they have holodeck technology that allows them to recreate Ancient Rome, but everybody talks to each other on television monitors. It's just so lame. The viewscreens have been around for over thirty years. Can't we move to something a little more interesting? But it's like pulling teeth." (Star Trek: Deep Space Nine Companion, p. 419)
Ira Steven Behr was completely behind Moore's idea; "Viewscreen scenes are always difficult to pull off. The longer they are, the more boring they are, and having a character talk to someone on a viewscreen is very distancing. And it did work in this episode. We never could have had Eddington on the viewscreen for all of his scenes. It would have been dramatic death." (Star Trek: Deep Space Nine Companion, p. 419)
Despite this, however, the holo-communicator was not seen as successful, something alluded to by Gary Hutzel, "It was a terrible idea from the get-go. The idea was to create this amazing 3-D image, but TV's a 2-D medium, so it's hard to show that it's 3-D. So you have to move the camera around so that audience can see that it's 3-D, but then it could look to them like the guy beamed in. So you have to find a way to deal with that. It created all these problems that the writers hadn't thought about, and it missed the whole point of why Gene Roddenberry wanted a viewscreen: so you could avoid unnecessary expense." (Star Trek: Deep Space Nine Companion, p. 420)
It would seem that the use of the holo-communicator was eventually discontinued by Starfleet, as it never appeared after "Doctor Bashir, I Presume". Ronald D. Moore suggested that it was moved into the Defiant's ready room for Season six. (AOL chat, 1997) As for the production reason, he stated that the writers never had a "viewscreen-type scene" that would be more effective with the holo-communicator. (AOL chat, 1998)
The holo-communicator was re-intruduced in Star Trek: Discovery, used frequently, albeit with more primitive graphics. Recalling how the idea of using holo-communicators so often in the series was decided on, Aaron Harberts characterized them as one of multiple "just slightly different ways of making it feel a little more contemporary."  Much like the "later" 24th-century attempt to re-introduce the technology, it would seem that this iteration of the technology was discontinued as well, as nothing of the sort was seen in Star Trek: The Original Series.
One similar technology was visible in Star Trek VI: The Undiscovered Country, where Azetbur on Kronos One viewed the President of the Federation on what appeared to be a monochromatic holo-communicator. Also, the Cardassian viewscreens on Deep Space 9 projected the image horizontally between two separate devices, not on an embedded device such as on Federation starships, suggesting that holographic technology is at least involved in Cardassian viewscreen design. The Xindi used a similar device in the 22nd century.
In the novel Demons of Air and Darkness, Starfleet uses an advanced version of this system to hold a meeting of numerous starship and station captains across the Federation. Each captain entered their own holodeck or holosuite, which then transmitted their own image to every other holodeck and received similar transmissions in turn, making everyone appear to be in the same room despite being across the Federation from each other.
In the novel Articles of the Federation, press briefings from the Office of President of the United Federation of Planets were conducted via holo-communications as of 2380. These briefings had formerly been conducted in person, but increased security during the Dominion War prevented this from continuing. From the point of view of the participant, they were in a room full of reporters from all over the Federation without ever leaving their homeworld, home, or even office.