(covers information from several alternate timelines)
An assignment patch, or mission patch, was a patch or badge seen on uniforms, most notably on Earth, where their use began in the 20th century. The United Earth Starfleet and MACO personnel continued to use assignment patches to denote the ship, station, or other post to which an individual was assigned, as did the later Federation Starfleet.
In 20th and 21st century Earth, assignment patches were used on military uniforms and civilian uniforms alike. NASA patches from that era appeared in the Crash-n-Burn Bar in Bozeman, Montana, and at the 602 Club in San Francisco. (Star Trek: First Contact; ENT: "First Flight")
In 1944 of an alternate timeline, Alicia Travers noted that the patch on Jonathan Archer's shirt said "Enterprise" and surmised he was a sailor on the USS Enterprise aircraft carrier. She asked him how he wound up so far from the site of the ship's sinking, but he didn't elaborate. (ENT: "Storm Front")
In the early 22nd century, Starfleet personnel wore assignment patches on the sleeve of their left arm, in a similar style worn during the initial period of space flight on Earth. The uniform was later updated, adding the Starfleet patch to the right arm. (ENT: "These Are the Voyages...") The assignment patch of Enterprise NX-01 was also used as a screensaver on monitors aboard the ship. (ENT: "The Catwalk")
In the mirror universe, the Terran Empire used the traditional styling from the 22nd century. While the Empire's uniforms had several variations that distinguished them from those of the prime universe, they still wore mission patches on their shoulders. By the 23rd century, the mission patch had been replaced with the Imperial insignia. (ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II"; TOS: "Mirror, Mirror")
By 2233 assignment patches had seemingly been done away with, although a badge in the shape of the Starfleet insignia placed over the left breast was a part of the uniform, at least on the USS Kelvin. (Star Trek)
Some time before 2256, Starfleet adopted assignment patch-like variations in these badges, placing smaller symbols inside the Starfleet insignia, to indicate different branches or specialties. (TOS: "The Menagerie, Part I", "The Menagerie, Part II", DIS) This also occurred in the alternate reality created by Nero's destruction of the Kelvin. (Star Trek; Star Trek Into Darkness; Star Trek Beyond)
Later in the mid-23rd century, Starfleet resumed the tradition of using unique patch emblems for different assignments, albeit placing them over the left breast. These patches usually but not always included the same symbols for the different divisions. The USS Enterprise's assignment patch was simply the Starfleet insignia. (TOS; ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II")
Assignment patches thus fell into disuse, and Starfleet eventually supplanted them with combadges in the shape of the Starfleet insignia. For example, such combadges were worn by the crew of the Soyuz-class USS Bozeman in 2278 and the Ambassador-class USS Enterprise-C in 2344. (TNG: "Cause and Effect", "Yesterday's Enterprise")
20th and 21st century patches
22nd century patches
23rd century patches
24th century patches
|Starfleet uniform styles|
|By era:|| 2140s-Early 2160s • Mid 2160s • 2230s • Late 2230s-2250s • 2250s-Early 2270s • Mid 2270s •|
Late 2270s-2350s • 2350s-2370s • Late 2360s-Early 2370s • 2370s • Early 2380s • Mid 2380s • 2390s
|By type:|| Casual duty attire • Patient wear • Fictional uniforms|
Alternate & parallel uniforms • Alternate reality uniforms • Terran Empire uniforms
|Related topics:|| Assignment patch • Federation blue • MACO uniform and equipment • Starfleet decorations •|
Starfleet insignia • Starfleet ranks • Terran Empire ranks • Xenylon • Uniforms of the Alpha Quadrant
According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 48), the term "mission patch" was "not spoken in dialog, but was used in scene descriptions in 'Affliction', where it described both ship and agency emblems. The classic Starfleet arrowhead emblem was created by costume designer William Ware Theiss."
The assignment patch for the USS Enterprise (NCC-1701) and the movie-era Starfleet assignment badge are very similar to those used by the US Space Command in the 20th century and early 21st century.
The assignment patch for Enterprise NX-01 was created by Wendy Drapanas, whereas the patch for the ECS Horizon was designed by Anthony Fredrickson. "When Wendy and I had our assignments on the pilot ['Broken Bow']," recalled Fredrickson, "one of hers was to design the NX-01 patch – which was a cherry assignment that I was very jealous she got! So when Mike [Okuda] gave me the Horizon patch, even though it was only for one show, I was very happy." (Star Trek: Communicator issue 144, p. 36)
Michael Okuda has also designed two real world emblems; the STS-125 mission patch in 2009, and more recently, the emblem for Johnson Space Center's Flight Operations. 
A memo from Robert Justman dated 18 December 1967 discusses the Starfleet arrowhead emblem and individual crew patches. It states "all Starship personnel wear the Starship emblem that we have established for our Enterprise Crew Members to wear," meaning that the arrowhead was always meant to be the emblem for all of Starfleet. Further, the crew of the Antares was the "equivalent of the Merchant Marines or freighter personnel" and other emblems should be counted as production mistakes.