Skagaran data modules contained data chips that looked extremely similar to crystals. Multiple data modules were on a Skagaran starship which, in the 19th century, crashed on a Skagaran colony planet. (ENT: "North Star")
Starfleet data modules consisted of several memory chips encased in a transparent coating and were similar to microtapes used in the 23rd century. Most of the control consoles aboard NX-class starships had slots for several data modules. (ENT: "The Andorian Incident")
On board the NX-class Enterprise NX-01 in 2151, Ensign Hoshi Sato used a data cube to digitally store a private message that the Vulcan cruiser Ti'Mur had covertly sent to the quarters of Enterprise's first officer, Sub-Commander T'Pol. After decrypting the message, Sato gave the data cube containing it to Commander Trip Tucker, explaining that the message was in Vulcan and that he would therefore have to process it through the ship's translation matrix to be able to understand it. Tucker then plugged the data cube into a slot on his desk in engineering, proceeding to do exactly as Sato had suggested. (ENT: "Breaking the Ice")
Later that year, Ensign Sato delivered a data module, containing a letter, to Doctor Phlox in the sickbay on board Enterprise. Although Sato initially suspected the letter might be from a female, Phlox explained to her it was actually from Dr. Jeremy Lucas, the first Human to ever serve on Denobula. (ENT: "Dear Doctor")
Aboard Enterprise in circa 2152, Commander Tucker showed Captain Jonathan Archer a data module that contained a complete recording of a water polo final between Stanford University and the University of Texas. Tucker had received the recording by subspace before storing it on the data module. He and Archer later used the storage device to watch the game on a monitor in the captain's mess. (ENT: "Vox Sola")
By 2153, several of the data modules in the wreckage of the crashed Skagaran ship were damaged. In that year, however, Tucker and T'Pol managed to access at least one of the craft's data modules and retrieve some data chips from it. While they were trying to interpret information from the data chips, Captain Archer asked them if they were having any luck with the data modules. The Enterprise crewmembers were indeed able to access logs from the data modules, which Ensign Sato then worked on translating. (ENT: "North Star")
In 2154, Captain Erika Hernandez, commanding officer of Columbia, gave Captain Archer a data module containing Hernandez's senior officer candidates, as she wanted Archer to give her his opinion of them. (ENT: "Home")
A computer module that Vulcan Ambassador Soval once referred to as a "Starfleet data module" showed its user gaps in Vulcan satellite surveillance, when beam-ins to the planet's surface couldn't be detected by the Vulcans. The precise moment when Soval called it a "Starfleet data module" was immediately after he gave it to Commander Tucker, when Soval feigned ignorance of the device's significance, as its transfer from him to the Starfleet officers wasn't officially permitted. (ENT: "The Forge")
In 2155, while Lieutenant Malcolm Reed was attempting to help ensure an away team from Enterprise managed to travel from the ship to the surface of Mars without being detected by John Frederick Paxton (who had hacked into the planetary sensor grid), a data module was given to Reed by Section 31 operative Harris, who Reed consulted and met with, one night in a San Francisco alley. Reed plugged the device into the back of a PADD he had with him; the data module showed him that, if the away team kept their altitude less than ten meters above the surface of Mars, they wouldn't be noticed by Paxton. (ENT: "Terra Prime")
Background information Edit
In the case of the Starfleet data modules, their name was not established on screen but comes from the final draft script of "Vox Sola". The scripts of "Breaking the Ice" and "Dear Doctor" consistently used the term "data cube", however (using "cube" for short). In fact, both scripts described this kind of device in each of those episodes as "small", implying that there was at least a significantly larger version of this storage medium. A text commentary track about "Vox Sola", written by Michael and Denise Okuda, termed this type of device a "data card". According to the commentary, the clear part of a data module is the same size as the "microtape" props used in Star Trek: The Original Series.
A Vulcan data module was to have been shown in ENT: "Fallen Hero", as evidenced by that episode's final draft script, though it isn't in the installment. The data module would have stored vital evidence that Vulcan ambassador V'Lar had gathered regarding corrupt Mazarites, proof she needed in order to testify against them. She said about the data module, "It contains all my notes, my testimony... months of work." The module was handed from V'Lar to Archer with instructions, if she was killed by the Mazarites, to deliver the unit to the Vulcan High Command.
The final draft script of "Singularity" commented that, upon leaving her science station while Commander Tucker was noisily tweaking the command chair, T'Pol extracted "a few data modules from their slots on her console."  In the final edit of the episode, though, it is difficult to see whether she does this.
The data module in "Home" was scripted (in the post-production, archival version of the final draft) to be a "data disc" instead. This was evidently changed, with a data module appearing on-screen. Similarly, the final draft script of "Terra Prime" (again, a post-production version) called for a "data chip" to be used, even though the final version of that device was clearly an identical prop to that used in "Dear Doctor", "Vox Sola", and "Home".