Gotei 13 General Information.

The Gotei 13 (護廷十三隊, Goteijūsantai; 13 Division Imperial Guards, 13 Court Guard Squads in the English dub) is the organization which most Shinigami join, and one of the three main branches of military in Soul Society. It is split into thirteen divisions (hence the name), with several divisions having certain specializations. The Gotei 13 is subservient to the edicts of the Central 46 Chambers. They also have no jurisdiction over any of the noble families. The symbol of the Gotei 13 is a rhombus, all the 13 Divisions symbols are placed within the rhombus on the back of the Captains haori. Officially, neither the Kidō Corps and Onmitsukidō are not part of the Gotei 13. However, as Soifon is both the Captain of the 2nd Division and the head of the Onmitsukidō, the Onmitsukidō is effectively under Gotei 13 control. In addition, the Captain of the Kidō Corps has been known to obey summons and orders from the the Captain-commander of the Gotei 13.
General Operation
The Gotei 13, in contrast to the normal operations of modern-day militaries, does not operate as a cohesive unit normally. The command of the divisions are generally left to the discretion of its particular Captain, who commonly runs it whatever way he/she sees fit, allowing for the change of a division's traditional duties to an entirely new set and even the creation of a separate institution connected to the division. The Gotei 13's main duty is to defend the Seireitei but they are also allocated districts Rukongai to defend.In addition to these duties, Shinigami also have to deal with the business of being assigned to the Human World.
The only general authority to call forth a combined front would be that of an edict from the Central 46 Chambers, or more often the Captain-Commander. It is not uncommon for Captains to not see each other as a entire group for long periods of time, except for cases involving a threat to all of the Soul Society, a Captain's punishment, or to a lesser degree, a Captain's promotion. On a regular bases, the Captains are spread out all across the Soul Society doing their divisions' duties, whatever they may be. They are, in fact, so far spread out that getting the Captains together as a group, or even getting their lieutenants together as a group, could normally take a day to accomplish.
Unless specifically ordered by the Captain-Commander, Captains can choose whether or not to attend meetings (at their own risk) or not even involve themselves in the daily situations of the Soul Society. Captains are not allowed, without permission, to carry out activities outside of their normal jurisdiction or their division. The Captain of one division can punish the subordinates of another Captain if warranted.

Size and Recruitment
Each Division is made up of just over 200 Shinigami, with the Gotei 13's standing force being just around 3000 total enlisted troops. The Shinigami are typically recruited through the Shinigami Academy, however, it must be noted that not all graduate make it into the Gotei 13. Graduates from the Academy are required to sit an entrance exam for the Gotei 13 and only those who pass are allowed to join. The entrance exam can be taken multiple times.


Ranks
As with any army, the Gotei 13 has ranks. There are 20 ranks for officers but no ranks for non-officers. The lower the number is, the higher the rank. The ranks and their descriptions are below, from highest to lowest in terms of strength.

Captain-Commander
The Captain-Commander (総隊長, Sōtaichō; Captain-Commander, Head-Captain in the English dub) is the leader of the Gotei 13 and currently the Captain of the 1st Division. The current holder of this title is Genryūsai Shigekuni Yamamoto.

Captain
A Captain (隊長, Taichō; lit. Unit Commander) is the leader of one of the thirteen divisions (with the special case of the Captain-Commander).
Captains are generally the most respected Shinigami within the Soul Society, with the Kidō Corps, and Onmitsukidō leaders possibly being on equal status. With one exception, all captains are able to utilize the final stage of their Zanpakutō, and are more powerful than even their lieutenants. Knowing Bankai gives captains an average of five-to-ten times the strength of Shinigami who are unable to perform it.They have also extensively-trained, if not mastered, their Bankai, allowing them to actually use the enormous power effectively in battle. Most captains are also skilled in the use of Flash Steps, Kidō, and generally have extensive knowledge of Shinigami history and battle tactics. All of these factors create a huge power-gap between captains and lower-ranked officers.
It is not commonly stated but in truth the power of the Soul Society lies in the Captains of the Gotei 13, as they are the leaders of its military arm. With them at full strength Soul Society is safe and near unbeatable, if they are broken their strength dwindles making them highly vulnerable.
There are three different ways to become a Gotei 13 captain:
Captain proficiency test (隊首, taishu): A test which requires the ability to perform Bankai. Presumably, most Shinigami become captains using this method. At least three existing captains, including the Captain-Commander, have to witness the test.
Personal Recommendation: To have personal recommendations from at least six captains and approval from at least three of the remaining seven.
Trial by Combat: To defeat a captain one-on-one with at least 200 witnesses from the captain's division. Kenpachi Zaraki is the only known current captain to have achieved his rank using this method. This method is seen as being the exceedingly rarer of the three as well as the least refined. The method is commonly looked down upon by some captains as barbaric compared to the other refined methods. This method also allows one to bypass having the mastery of bankai or other possible abilities that a captain would otherwise be required to have knowledge of, as it allows no judgement from captains who would require those skills as a prerequisite to join their ranks.
Lieutenant
A Lieutenant (副隊長, fukutaichō; Vice-unit Commander, often translated as Vice-Captain) is the 2nd seated officer in a division.
Lieutenants function as executive officers for their division, taking care of or supervising day-to-day operations. They (along with their captains) are not assigned to one specific squad. They generally only know the Shikai of their Zanpakutō, but are still the second strongest in their division. In the case of a division captain's death, departure, or other circumstances making him unable to perform their duties, the lieutenant acts as the "substitute captain" until another can be assigned.
Seated Officers
Seated officers are the remainder of the officers in the Gotei 13 that hold ranks at or below 3rd Seat. Captains and lieutenants have the ranks of 1st and 2nd Seats, respectively. Next to that, there are 18 more seats (3rd-to-20th). These ranks are somewhat analogous to junior officers or non-commissioned officers in modern military. There seems to be a distinction between senior and junior seated officers, just like in real-life armies. In the Gotei 13, only one person can hold a senior officer rank (such as 3rd or 5th seat), but there are several holders of junior officer ranks (e.g. 20th seat). This may vary from division to division however, as there are two 3rd Seats in the 13th Division. Seated officers often lead subunits within each division (for example, Hanatarō Yamada is leader of 4th Division's 14th Advanced Relief Team).
Gate Guardians
The Gate Guardians are the guardians of the four gates of Seireitei. They are all giants who have the strength to lift the gates made of Sekkiseki.
Jidanbō Ikkanzaka - The keeper of the West Gate.
Hikonyūtō - The keeper of the South Gate.
Danzōmaru - The keeper of the North Gate.
Kaiwan - The keeper of the East Gate.

Others
This is a list of unseated Shinigami and (if known) their respective divisions.
Rikichi - Shinigami of the 6th Division
Rin Tsubokura - Shinigami of the 12th Division.
Hiyosu - Shinigami of the 12th Division.
Akon - Shinigami of the 12th Division.
Rukia Kuchiki - Unseated Shinigami of the 13th Division due to her brother's influence, but powerful enough to qualify as a seated officer
Zennosuke Kurumadani - Rukia's replacement as watcher of Karakura Town, his Division number and rank is unknown.
Ashido Kanō - A Shinigami that was stranded in Menos Forest of Hueco Mundo (Anime-Only)
Senna - A character in Bleach: Memories of Nobody
Eikichirō Saidō - A Shinigami sent to find Rukia (Anime-Only)
Ran'Tao - Leader of the research team that created the Bounts (Anime-Only)
Baishin - A former Shinigami and the antagonist in the OVA The Sealed Sword Frenzy
Ganryū - Leader of The Dark Ones in Bleach: Memories of Nobody (Deceased)
Sōjirō Kusaka - The main antagonist in Bleach: The DiamondDust Rebellion (Deceased)
Matsuri Kudō - A Shinigami of the 5th Division (Video game-Only)
Fujimaru Kudō - A Shinigami of the 5th Division (Video game-Only)

Division Speciality
It has been shown that the 13 divisions and the Kidō Corps may each specialize in a certain area, but only a few division and their respective specialities have been made known.