A captain is generally the leader of a single ship. However, when a crew is large enough to have multiple ships—also known as a fleet—each ship may be commanded by a single captain (or division commander), and the entire fleet is controlled by an admiral, whom the captains answer to. The admiral has absolute command over the entire fleet of ships, having only captains to serve as second-in-commands, dividing their numbers into multiple ships and divisions, making it easier to control the organization.
While there are several pirate crews with large numbers of ships and sailors on board, not all of them necessarily designate their overall captain as an "admiral", instead remaining as "captain" with division leaders known as "commanders." Such crews include the Whitebeard Pirates and Big Mom Pirates.
Krieg is the first pirate admiral to be introduced. He was in command of the Krieg Pirates, also known as the Pirate Armada. In command of 50 ships and 5000 pirate soldiers, Krieg was considered the most powerful pirate in East Blue in terms of military power. Contrary to the crew's fearsome reputation, however, Krieg dominates them with an iron fist, using fear to keep them under his thumb, thus severely limiting their true strength.
Mont Blanc Noland was an admiral of his ship from Lvneel Kingdom. He was an explorer from 400 years ago, having made multiple trips throughout the world and made many discoveries that were doubted by many. He was, in truth, a very caring person who would go to great lengths for the well-being of his crew.
Shiki is the admiral of the Golden Lion Pirates, although his crew universally referred to him as oyabun. He commanded a fleet of at least 25 ships when facing the Roger Pirates at Edd War. Much of this crew was decimated from the battle. After escaping from Impel Down, Shiki went to rebuild his fleet in an attempt to conquer the world, gaining many allies from other pirates.
Marshall D. Teach was originally a captain of his own crew, the Blackbeard Pirates, which only had a maximum of eleven members in total. After usurping the position of Yonko from the late Edward Newgate, the Blackbeard Pirates grew to a fleet of ten ships, each led by one of the Ten Titanic Captains, and Teach himself became the admiral; first ship captain Jesus Burgess continues to refer him as "Captain Teach", which fifth ship captain Laffitte reprimands him for.
Chinjao is the 12th Don of the Happo Navy, a pirate crew which consisted of 1000 men. After losing his "key" to the treasure under the Jewel Ice Sheet to Monkey D. Garp, he retired from the position with a broken heart. His grandson, Sai, succeeded him as the 13th Don of the Happo Navy. Under Sai, he integrated the Happo Navy as 3rd division of the Straw Hat Grand Fleet.
A Marine Admiral is referred as a "Taisho" (大将,Taishō?), which differs from a pirate admiral, referred as "Teitoku" (提督,Teitoku?).
Teitoku can also translate as "Commodore". Once again, this differs from the Commodore of the marines, which is translated from "Junsho" (准将,Junshō?).
Don (首領,Don?) is honorific used for people of long-standing leadership of a community, primarily used in Spain, Portugal, Italy, the Americas and the Philippines. It can also refer to a crime boss, especially the mafia.
Although never stated, it is implied that Z's position is mentioned with the english pronunciation (アドミラル, Adomiraru) because of his direct subordinate, Ain, refering herself as Vice-Admiral in the same way (バイスアドミラル, Baisu Adomiraru).
Nelson Royale, an anime-exclusive Marine officer, held the rank of Commodore, yet was referred as "Teitoku" (提督,Teitoku?) instead of "Junsho" (准将,Junshō?), a rank that did not exist in the Marine ranks. This was apparently an oversight of the anime producers at the time, since the author did not design the corresponding arc.
With the formation of the Straw Hat Grand Fleet, Luffy can be considered a pirate admiral in a sense; however, Luffy himself did not personally agree to take the representatives on as members of his crew, but rather the representatives swore their loyalty to the Straw Hat flag by their own free will, more similar to an alliance.