(covers information from several alternate timelines)
Starfleet Medical was a department of (United Earth) Starfleet dedicated to biological research and treatment of its personnel. Based out of San Francisco on Earth, Starfleet Medical oversaw all Starfleet medical personnel and facilities, which included exercising co-authority, besides the captain, over starship sickbay personnel.
As its emblem, the department carried, from the mid 2150s onward, a stylized version of the double-snaked medical caduceus symbol from ancient Greek mythology. With the launch of the NX-class, Starfleet Medical asserted its co-authority over sickbay, by having its symbol prominently featured on the transparent access doors to sickbay (ENT: "Broken Bow"), something repeated two centuries later with the Galaxy-class. (TNG: "Remember Me") In the 2270s and the 2280's (when the emblem was shortly replaced with a variant design), the emblem was worn by starship medical personnel on their garments, when on duty in sickbay. (Star Trek: The Motion Picture, Star Trek II: The Wrath of Khan)
In an alternate timeline, Starfleet Medical was known to operate at least one hospital ship of its own, the USS Pasteur, as indicated by its emblems on the primary hull and their prominent presence on the bridge. (TNG: "All Good Things...")
In the 23rd century, an office existed known as the Starfleet Surgeon General which held some command authority over Starfleet Medical, including the power to relieve starship medical officers for incompetence. (TOS: "Turnabout Intruder")
It was also responsible for all medical research activities in the Federation. Researchers were required to obtain approval from Starfleet Medical before proceeding with new treatments and medications. (TNG: "Ethics")
In 2151, the Klingon courier Klaang was brought to Starfleet Medical after being shot with a plasma rifle while on Earth. There, he was treated by Doctor Phlox, who had been stationed at Starfleet Medical through the Interspecies Medical Exchange. Captain Archer subsequently asked Phlox to join his crew as the chief medical officer aboard Enterprise NX-01. (ENT: "Broken Bow")
In 2369, Doctor Julian Bashir told a Bajoran woman he dated in the Replimat about his exam at Starfleet Medical. (DS9: "Q-Less") Shortly thereafter, he told Major Kira Nerys that he learned, in his first year at the medical school, never to trust a tricorder. (DS9: "The Passenger")
Odo visited Starfleet Medical in 2372, where they performed medical tests on him to gain information that could be used against the Founders. It was during this examination that Odo was infected with a morphogenic virus by the covert intelligence group Section 31. (DS9: "When It Rains...")
Odo contracted the morphogenic virus in late 2375, a disease that had already begun killing the Founders of the Dominion. Starfleet Medical was unwilling to assist Julian Bashir in his research for a cure and refused him access to Odo's medical file, believing Bashir was trying to help the enemy. When Benjamin Sisko requested the file, they sent a copy of Mora Pol's medical file on Odo, taken decades previously, in an attempt to hinder Bashir's efforts. (DS9: "When It Rains...")
In an alternate timeline, in which it took the USS Voyager twenty-three years to return to Earth, Tuvok was sent to live at Starfleet Medical, having succumbed to a mental condition while aboard in 2378. The holographic doctor of Voyager was working at the medical facility in 2404 and was helping Tuvok, as well as working on the project to develop chronexaline. (VOY: "Endgame")
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The stylized medical caduceus symbol was designed by Lee Cole, the graphic designer for Star Trek: The Motion Picture and was seen as signage in sickbay and more prominently as badges on garments of medical personnel on duty in sickbay. For Star Trek II: The Wrath of Khan, Cole came up with a more artful variant which was now worn as a pin instead as a badge, and was only featured in this production. Cole's successor for the modern television franchise, Scenic Artist Mike Okuda, adopted her original design, and the symbol was most frequently seen throughout the run of the modern television franchise on the new medkits, introduced at the start of Star Trek: The Next Generation' s fourth season. (Star Trek: The Next Generation - The Continuing Mission, p. 117) Okuda also retro-applied the symbol as signage throughout the sickbay of the Enterprise NX-01 in Star Trek: Enterprise, thereby canonically establishing that the symbol had been in use for over two centuries. (ENT: "Broken Bow", "Dear Doctor")
Yet, both Cole and Okuda, unwittingly perhaps, perpetuated the misconception (especially held in the United States) that the double serpent-entwined caduceus was the proper symbol for medicine and healing, which it was not. The proper symbol has traditionally been the Rod of Asclepius, a single serpent-entwined rod wielded by Asclepius, in Greek mythology the god of healing and medicine. The caduceus on the other hand, was the rod carried by Hermes, the god of travelers, commerce and thieves, and his rod has since then become the symbol of commerce. For unclear reasons, the misconception originated in the United States when the caduceus was introduced shortly before the American Civil War as the symbol for the US Medical Corps. Surviving Civil War uniforms of the Union Medical Corps show the symbol embroided on a green band, worn as a chevron on both sleeves. (Echoes of Glory: Arms and Equipment of the Union, Time-Life Books, 1991, pp. 158-159) Incidentally, the color green was adhered to in the first six Star Trek films as the color signifying medical divisions, shown in the fields of their personnel's insignia as well as being the color of their undershirts, before returning to the various shades of blue, the generic color signifying all sciences divisions, as already established in Star Trek: The Original Series. The misconception was perpetuated for nearly a century in the US armed forces until the interbellum years, when the medical branches of the US Airforce  and the US Navy  started to carry the proper symbol on their coat of arms. The US Army followed suit, but still uses the caduceus for their individual (regimental) badges and insignia.