(covers information from several alternate timelines)
The Starfleet insignia was an asymmetrical, arrowhead-shaped pennant adopted by Starfleet as its identifying emblems. This delta was displayed on starship hulls, installations, uniforms worn by Starfleet personnel, and equipment used by that personnel.
The United Earth Starfleet of the 22nd century used a simple forward-facing "pointer" shaped pennant, with a round backdrop, that was reminiscent of symbols used by NASA in the 20th and 21st centuries. The original design for the Starfleet uniform of that era did not include this symbol, unless it was a crewperson assigned to Starfleet Command, who wore the insignia as their assignment patch on the left shoulder. In a later uniform revision of the 2160s, personnel wore the Starfleet patch on their right shoulder as well as the left shoulder assignment patch. (ENT: "First Flight", "These Are the Voyages..."; TNG: "The Royale")
Shortly after the founding of the United Federation of Planets in 2161, a solid silver delta was worn on the left breast of the uniform, with individual assignment patches worn on both shoulders. (Star Trek Beyond) This simple silver design would see a brief resurgence in the early-2380s. (LD: "Second Contact")
In the mid 23rd century, a simplified pennant version of the delta was used on Federation starships and installations. This symbol for Starfleet was not worn on uniforms, which again only displayed an individual crewperson's assignment patch. Some starships, such as the Defiant, had an assignment patch similar to this symbol. (TOS: "Court Martial"; ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II")
The delta insignia came into usage for much of Starfleet's exploration division in the early to mid 23rd century. A starburst insignia was used by flag officers and administrative support staff of fleet headquarters and starbases. Some Constitution-class starships during the mid-23rd century used a variety of insignia unique to each ship. The Enterprise maintained the delta as her insignia and assignment patch. (TOS: "The Cage", "Court Martial", "The Doomsday Machine", "The Omega Glory"; ENT: "In a Mirror, Darkly, Part II"; Star Trek; DIS: "The Vulcan Hello")
By the mid 2270s, while installations and starships, such as Epsilon IX station, maintained their individual assignment patches, Starfleet Command had adopted a new Starfleet insignia, a symbol previously used as the assignment patch of the USS Enterprise, and prior to that, the USS Kelvin and USS Franklin. (Star Trek: The Motion Picture; Star Trek; Star Trek Beyond) Specifically, the command division insignia, with the MACO five-pointed star contained in the arrowhead, was most often used, although a version omitting the internal star gained prominence as well. This arrowhead shape, with similar asymmetry as the previous incarnation, had roots in history as being used in a basic form on the United States armed forces space command ratings badge and unit insignia, and on UESPA exploration vessels, such as the Friendship 1, in the 21st century. This symbol was now pointed up when worn on the chest or shoulder of uniforms, as well as displayed in signage. The arrowhead was faced forward when used as hull decoration. This symbol eventually phased out all assignment patches by the late 2270s, and remained in use for some Starfleet divisions well into the 24th century. The metal badge version of this symbol became Starfleet's first combadge as well. (Star Trek II: The Wrath of Khan; TNG: "Yesterday's Enterprise"; VOY: "Friendship One")
In the 2340s, a more stylized version of the symbol, the familiar arrowhead without any internal symbology within the shape, and mounted on an oval field, was incorporated into uniforms. A metal badge of this emblem, again serving as a combadge, was worn on the uniform chest. This symbol also became common in starship hull pennants. (TNG: "Encounter at Farpoint", "Family")
By the 2370s, a newer version was created for use as a combadge, with a stylized trapezoidal background. Starship hull pennants of this era began using a simplified version of the arrowhead alone, without a background. (Star Trek Generations; DS9: "The Search, Part I"; VOY: "Caretaker"; Star Trek: Insurrection) Starfleet briefly adopted the simplified silver version as part of the uniform used in the early 2380s before switching back to the version used in the 2370s. (LD: "Second Contact"; PIC: "Nepenthe")
By 2399, a hollow, outlined version of the arrowhead in silver with two dark, irregular stylized vertical trapezoids descending from behind was the Starfleet insignia. (PIC: "Maps and Legends") This is similar to a version of the insignia used in several alternate timelines during the same time period, which used gold. (TNG: "All Good Things..."; DS9: "The Visitor"; VOY: "Timeless", "Endgame")
In the alternate reality created by Nero's incursion, Starfleet had done away with the assignment patches by the 2250s. Instead the fleet was united behind a gold arrowhead, slightly modified from the assignment patch of the USS Kelvin, with a shallow arrowhead indentation on the inside (as seen on dress uniforms). The insignia used on starships was a silver arrowhead with a hollow division symbol inside the arrowhead, similar to the assignment patches used aboard the prime timeline's USS Enterprise. (Star Trek)
In the mirror universe, a dagger impaling the Earth was the symbol of the Terran Empire and as such was used as the insignia of the Imperial Starfleet. (ENT: "In a Mirror, Darkly", "In a Mirror, Darkly, Part II"; DIS: "Despite Yourself"; TOS: "Mirror, Mirror")
Parallel and future versions
In the Barash-created false-future illusion experienced by William T. Riker in 2367, the symbol used in the Starfleet combadge consisted of the arrowhead, either silver or gold, with four horizontal bars behind indicating the rank of the individual by the number of bars that were gold or silver. (TNG: "Future Imperfect") This version was also in use in an alternate quantum reality visited by Worf in 2370. (TNG: "Parallels")
In several alternate timelines observed by Starfleet personnel, new variations of this symbol existed. In the 2390s and 25th century, Starfleet personnel were known to wear a hollow, outlined version of the arrowhead with two irregular stylized vertical trapezoids behind it as their badge. (TNG: "All Good Things..."; DS9: "The Visitor"; VOY: "Timeless", "Endgame")
Further in the future, 29th century Starfleet had a rotated version of the arrowhead shape, with the bottom half filled in with a mirror of the top point of the shape. In 29th century timeship hull decorations, the arrowhead was shown pointing backwards from the direction of motion for the first time, possibly because of the vessels' capability for time travel. (VOY: "Future's End", "Future's End, Part II", "Relativity")
|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
From the Star Trek Encyclopedia (4th ed., vol. 1, p. 365), "The distinctive arrowhead symbol used on Starfleet uniforms was first created by Original Series costume designer William Ware Theiss for the "The Cage" in 1964. Three versions of the original symbol were created, used for command personnel, science specialists, and engineering staff. (A fourth version, featuring a red cross, was occasionally worn by Christine Chapel.) During the original Star Trek series, it was generally assumed that the arrowhead symbol was unique to Enterprise, and that other starships had different insignia for their uniforms."
Another version, which appeared only in "The Cage", had a unique symbol of a box, open on the wearer's left, with a "C" shape inside. Only two crewmembers were seen wearing this unique insignia; both were on the bridge with one standing like a guard at the turbolift doors, and the other standing at times next to Captain Pike's chair. This unique insignia's department/division was never mentioned on screen; it was dropped in the second pilot, "Where No Man Has Gone Before", and never appeared again.
A memo from Bob 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. 
Reference works, such as the Star Trek: The Next Generation Technical Manual (p. 3), suggest that the Starfleet-wide adoption of the Enterprise emblem was probably to honor the ship and her crew for their successful and historic five-year mission. Considering that one of Star Trek's long-held influences is the Age of Sail, it may be a reference to the British Royal Navy's adoption of Nelson Chequer – the particular coloration of the ships of Admiral Horatio Nelson – on all of its ships after the Battle of Trafalgar. In a form of fact emulating fiction (and vice-versa), the US Air Force Space Command badges have a delta arrowhead very similar to the 1960s-designed Enterprise insignia, while the United Earth Starfleet insignia seems to be a deliberate synthesis between the TOS era pennant and the NASA logo. Several other space agencies (Roscosmos, JAXA, CNSA, ISRO) use a variation of the 'vector' shape in their logos as well.
The Star Trek Encyclopedia further notes: "This changed in Star Trek: The Motion Picture, when a modified emblem, designed by Robert Fletcher, was used not only on Enterprise crew members, but on all Starfleet personnel. We therefore assume that at some point after the original Star Trek series, the Enterprise emblem was adopted for the entire Starfleet. The feature film insignia (in a couple of variations) was used for the movies set in the Kirk era, as well for Star Trek: The Next Generation flashback sequences involving Picard's cadet days. Yet another variation was created for Star Trek: The Next Generation's first season by Theiss, in conjunction with Rick Sternbach and Mike Okuda. This version was also used on Star Trek: Deep Space Nine. Most recently. Starfleet officers wear a version designed by John Eaves and Bob Blackman, first seen in Star Trek Generations. Additionally, two hypothetical future versions have been seen. One, designed by Okuda, was seen in "Future Imperfect" and "Parallels", while another, designed by Eaves, was used in "All Good Things..." and "The Visitor". Still more variants were added for the Kelvin timeline for Star Trek and Star Trek Into Darkness."
Following the announcement of the logo of the United States Space Force, a division of the Department of the Air Force, a number of sources noted its similarity to the Starfleet insignia, though others pointed out that the design appeared to based on the logo of the Air Force Space Command, which was phased out in place of the new Space Force.