(covers information from several alternate timelines)
The Starfleet tricorder was the versatile, portable sensing device developed by Starfleet R&D specifically for use by Federation Starfleet personnel. Tricorders were outfitted with the latest cutting-edge sensory and emitter technology.
In service aboard the early 2230s ship the USS Kelvin, the tricorder was a large flat device resembling a briefcase, complete with a handle for carrying the device. In the alternate reality, one was used to monitor the heavily pregnant Winona Kirk's health during the Narada's attack. (Star Trek)
In service during the mid-2260s, the standard Starfleet tricorder incorporated a segmented design. This tricorder, a generally square-shaped device, utilized three sections: a pivoting upper portion containing the display readout and controls; a compartmented mid-section containing data chips; and an adaptable lower portion, sometimes containing a hand-held sensor as with the medical tricorder. The tricorder of the 2260s was mostly black with silver trim and featured a leather-like strap, allowing one to sling the tricorder over the shoulder when not in use. (TOS: "The Naked Time")
Yeomen serving aboard Starfleet starships (such as Janice Rand aboard the USS Enterprise) often carried tricorders in order to record and retrieve information for the starship captain. More often however, the tricorder was standard equipment included on away missions. (TOS: "The Man Trap", "Charlie X")
A heavier version of the tricorder was also used by landing parties and aboard ship. This configuration sported a pistol-grip and its controls were located on the top of the unit. It was used primarily to detect contamination caused by ores and gases, as well as to register radiation and atmospheric pressure. (TOS: "The Naked Time", "The Enemy Within", "The Doomsday Machine", "Obsession")
With the widespread redesign of many facets of Starfleet and Federation technology in the early 2270s, so too did the tricorder advance toward a new design.
Smaller than that of the 2260s, the new hand-held Starfleet tricorder was blue-gray in color and featured two grips on either side of the body. Controls and readout displays were located beneath a lid at the top of the device. (Star Trek: The Motion Picture)
Among the options was a heavy-duty tricorder, often used during landing parties. Bulkier than previous designs, this tricorder featured a handle and a large head featuring a line of blinking indicator lights. (Star Trek II: The Wrath of Khan)
Harking back to the designs of the 2260s, a second tricorder option in use during the mid-2280s featured a familiar square, black and silver design.
Much smaller than the heavy-duty version, this model featured a slide-up top studded with controls and indicator lights. Also incorporating a strap, the tricorder could be slung over the shoulder when not in use. Unlike its predecessors, this tricorder possessed control surfaces over much of the body of the instrument. (Star Trek III: The Search for Spock)
By the early 2360s, the Starfleet tricorder had taken on a more streamlined appearance, distinguished by a flip-open design maximizing both portability and interface surface availability.
Incorporating several sensor clusters, these tricorders featured multi-channel communications assemblies, and multiple databank modules. Encompassing visual displays and graphic touch pad interfaces, the tricorder of the 2360s provided easy means for on the spot archive retrieval, the recording of away mission events, and constant scientific measurements including biological, geographical or meteorological.
TR-560 Tricorder VIEdit
In service as early as 2361, the TR-560 Tricorder VI model sensing device was available to officers serving aboard such vessels as the USS Victory and the USS Enterprise-D. (TNG: "Identity Crisis", "Encounter at Farpoint") With an array of sensors on the front of the device, this tricorder design flipped open to reveal a large number of control surfaces and a square display screen, roughly the same size as those utilized as early as the 2260s.
The TR-560 tricorder was capable of communicating with non-Starfleet computer systems and contained a built-in universal translator. Data used one of these tricorders to access and replay the logs of the captain of a Promellian battle cruiser discovered by the Enterprise in 2366. (TNG: "Booby Trap")
The tricorder was vulnerable to some computer viruses, even capable of transmitting them to other systems. A tricorder used by Data while investigating the Iconian homeworld fell victim to one of their viruses. In order to prevent propagation of the virus, Captain Jean-Luc Picard ordered the tricorder destroyed by Worf, who did so using a phaser. (TNG: "Contagion")
This version of the tricorder was not normally set up to detect neutrinos, nor was it typically designed to interface with a VISOR. Nevertheless, Lieutenant Commander Geordi La Forge was able to jury rig a tricorder to communicate with his VISOR and provide a bearing on a neutrino source when he was stranded on Galorndon Core in 2366. (TNG: "The Enemy")
See medical tricorder.
TR-580 Tricorder VIIEdit
In service between 2366 and 2372, the Starfleet TR-580 Tricorder VII succeeded the TR-560 Tricorder VI. Retaining the same basic shape as the previous design, the TR-580s also incorporated hand sensors, but this feature was eventually phased out by 2370. (TNG: "The Vengeance Factor"; DS9: "The Way of the Warrior")
TR-580s could be used for a variety of tasks, including recording time trials. (TNG: "Chain of Command, Part I") This tricorder was also configured to perform proximity checks every two minutes (DS9: "Move Along Home") and was able to emit a photoplasmic trail to help locate lost crewmembers. (VOY: "Twisted")
To prevent interference with objects, the TR-580 could be set to perform a passive scan. The model could also be locked into a continuous scan, thereby recording data, even when stowed. TNG: "Schisms" Capable of communicating with non-Starfleet computer systems, the tricorder could be linked to other units, acting as a small computer network, easily relaying information to other locations. (TNG: "Brothers"; VOY: "Emanations", "Heroes and Demons")
Sensor emissions from TR-580s could cause unpredictable changes in certain temporal distortions, and sometimes included polaric fields, allowing a user to scan for subspace fractures caused by polaric detonation. (TNG: "Timescape"; VOY: "Time and Again")
Starfleet sometimes distributed TR-580 model tricorders to Federation allied worlds, including Bajoran personnel on space station Deep Space 9. They could also be used by civilians, and were shipped en masse to areas in need, such as disaster sites. (TNG: "True Q"; DS9: "Babel", "Tears of the Prophets")
The standard TR-580 could function as a makeshift medical tricorder if necessary, although not all medical personnel were happy to do this. (TNG: "Time's Arrow, Part II"; DS9: "Emissary"; VOY: "Caretaker")
A variety of the standard tricorder appeared in USS Voyager's sickbay in the year 2371. Five tricorders of this type were included in little alcoves behind the three biobeds and to the left and right of the main entry to sickbay. (VOY: "Eye of the Needle") In 2372, these tricorders were replaced by PADDs. Similar tricorders were included in the same locations in sickbay aboard the USS Enterprise-E in 2373.
See medical tricorder.
TR-590 Tricorder XEdit
Succeeding the TR-580 Tricorder VII, the TR-590 Mark X tricorder was introduced sometime during 2371 (VOY: "Relativity", "Fury", "Shattered") and was used alongside the previous version until 2372. (VOY Season 1) It featured nearly identical control interfaces. The entire unit, however, was streamlined and made much smaller, featuring a more angular design. (DS9: "The Way of the Warrior"; VOY: "Initiations") These tricorders were stored in most equipment lockers on board the USS Voyager. (VOY: "Relativity")
The Mark X tricorder could still function after being underwater. (DS9: "Rocks and Shoals"; Star Trek: Insurrection) It could also function in hostile environments such as temperatures of 500 Kelvin, (VOY: "Demon") the upper atmosphere of a planet, (VOY: "Rise") and planets suffering nuclear winter. (VOY: "Friendship One")
The upper component of the TR-590 was capable of interfacing with a Starfleet desktop monitor and projecting a recorded image from the tricorder onto the screen. (VOY: "Timeless") The Mark X tricorder could also upload data it acquired remotely to a Federation starship. (VOY: "Initiations", "Future's End", "Year of Hell", "Equinox", "Good Shepherd")
While the USS Enterprise was in the year 2063, Captain Picard linked a Borg neural processor directly to a tricorder to learn of the Borg's plans to transform the deflector dish of the Enterprise into an interplexing beacon. (Star Trek: First Contact)
After an incident with the timeship Aeon threw Voyager back to 20th century Earth, Captain Kathryn Janeway used a tricorder of this type to to access secured files on a computer at Chronowerx Industries. (VOY: "Future's End")
The subspace waveguide in a tricorder of this model could be easily configured to generate a dispersion signal, preventing transporter lock. While under the influence of errant subroutines, the Doctor used a tricorder to prevent Voyager from locating him after he abducted Kes. (VOY: "Darkling")
Scans from this tricorder created an electrostatic cascade when interacting with charged methogenic particles. This would cause the particles to ignite. This occured when a Voyager away team boarded a damaged Malon export vessel flooded with methogenic particles, preventing use of tricorders during the mission. (VOY: "Juggernaut")
The Mark X tricorder was reportedly heavy for its size as indicated by B'Elanna Torres when she received a small gift from Tuvok during a holodeck generated baby shower. Torres asked if it was her child's first tricorder as the gift was small yet heavy. (VOY: "Human Error")
See medical tricorder.
TR-590 Tricorder XIEdit
Introduced as late as 2379, yet another iteration of Starfleet tricorder departed from the distinct, flip-open style preceding it.
Distinguished by a slim, PADD-like appearance, the tricorder in use aboard such Starfleet vessels as the Enterprise-E in the late 2370s featured a large touchscreen interface. Silver in color, it was trimmed with black pads at the bottom and sides, with several buttons at the top. These buttons could be covered by a small hinged door, lined with indicator lights.
Among common data-gathering tasks, this version tricorder could be used to interface with starship systems including force field control.
The medical tricorder of the same make was identical. (Star Trek Nemesis)
Tricorders have appeared in various alternate timelines.
In the alternate reality, the tricorder of the 2250s resembled its prime reality counterpart but was a smaller handheld device. Spock used the tricorder to activate a cold fusion device in the heart of a volcano on Nibiru in 2259. (Star Trek Into Darkness)
An extension of the Mark X model, this tricorder featured the familiar hinged cover design, but contained more touch control surfaces in place of toggle switches. The casing was black or dark-gray. (VOY: "Endgame")
During a possible version of the 29th century, the crew of the USS Relativity used a slim and sleek version of the tricorder, labeled a TR-890 Tricorder XV. Seven of Nine was given a TR-890 as she hunted for a temporal disruptor hidden on Voyager by Captain Braxton. (VOY: "Relativity")
In addition to the TR-890, a smaller, palm-sized device was in use during this period. Henry Starling found one aboard the Aeon and carried it with him at all times. He used it in an attempt to escape Voyager's transporter when the ship tried to retrieve the Aeon in 1996. (VOY: "Future's End, Part II")
Background information Edit
In 2012, Scanadu announced the SCOUT, which has been described as the real-life tricorder. The US$150 device can show one's temperature, respiratory rate, electrocardiography, blood pressure (systolic, diastolic), stress and oximetry levels by scanning a person's left temple. 
In the future 2400's timeline of Star Trek Online, tricorders are shown to have a more hinged design reminiscent of the Mark VI through X designs, including removable scanners. To perform various tasks, Starfleet, Klingon Defense Force and Romulan personnel all use tricorders in conjunction with other equipment (both worn and carried) capable of projecting "floating" holographic interfaces.