What Bankais exist within Soul Society? Bankais are the zanpakutōs' ultimate form. They have their own name and usually a special ability. Apparently, not everyone has a zanpakutō with this form. And it looks like bankais are exclusive to captains, since Ichigo can use this form as well.

So what are the zanpakutō with bankai form that have already appeared in the anime/manga (along with their names and abilities if possible)?
As SingerOfTheFall pointed in his answer Captain Byakuya stated that: 


  Even the four noble families, whose members are always exceptionally
  talented, have only a member capable of bankai every few generations.


At wikipedia states that:


  Only 4 non Captains in the Gotei 13 to have achieved Bankai


First Division

Genryūsai Shigekuni Yamamoto (山本元柳斎 重國 Yamamoto Genryūsai Shigekuni)


  Ryūjin Jakka's bankai "Zanka no Tachi" (残火の太刀?, lit. "Long Sword of
  the Remnant Flame") upon release causes all of the flames produced by
  Yamamoto to disappear into the blade. The bankai is comprised of four
  abilities based on the cardinal points:
  
  
  "Zanka no Tachi - Higashi (East): Kyokujitsujin" (残火の太刀･東　旭日刃) in which all the flames are concentrated on the edge of the blade,
  incinerating everything it touches
  "Zanka no Tachi - Nishi (West): Zanjitsu Gokui" (残火の太刀･西　残日獄衣) where Yamamoto envelops his whole body in flames, reaching up to
  15,000,000 degrees
  "Zanka no Tachi - Minami (South): Kakajuuman' Okushi Daisoujin" (残火の太刀･南　火火十万億死大葬陣) - this techique resurrects the burned and
  incinerated dead which were struck down by the blade using its burning
  state, it is roughly translated to "great army of the 10 trillion
  buried by flames".
  "Zanka no Tachi - Kita (North): Tenchi Kaijin" (残火の太刀・北 天地灰燼) - a powerful slash attack, incinerating everything in one fell swoop.
  Translated as "heaven and earth are burned to ashes".
  


Chōjirō Sasakibe (雀部 長次郎 Sasakibe Chōjirō)


  His bankai is "Kōkō Gonryō Rikyū" (黄煌厳霊離宮, lit. "Yellow-Glittering
  Royal Villa of the Stern Spirit") can produce a bolt of lightning from
  the blade which extends and transforms into a dome of lightning,
  stationed far above the users head and fastened to the ground by a
  large number of lightning pillars. These pillars can be controlled at
  will to shock the opponent.


Shunsui Sōzōsuke Jirō Kyōraku (京楽 次郎 総蔵佐 春水 Kyōraku no Jirō Sōzōsuke Shunsui)


  Unlike other Soul Reapers, Shunsui possesses paired zanpakutō called
  Katen Kyōkotsu (花天狂骨?, lit. "Bones of Heavenly Blooming Madness").
  While the blades are normally two daisho swords, their Shikai state
  resemble dao or falchions. When released with the command "Flower Wind
  Rage and Flower God Roar, Heavenly Wind Rage and Heavenly Demon Sneer"
  (花風紊れて花神啼き 天風紊れて天魔嗤う Hana Kaze Midarete Kashin Naki, Tenpū Midarete
  Tenma Warau), Katen Kyōkotsu gains the ability to "turn children's
  games into reality." One such game, Bushō Goma (不精独楽?, Lazy Spinning
  Top), involves spinning around wind like a top. Another, Takaoni (嶄鬼,
  Mountain Demon), declares that whoever is the highest is the winner. A
  third technique, Kageoni (影鬼, Shadow Demon), forces the players to
  alter their own shadows; whoever steps on a shadow, even their own, is
  declared the loser. The fourth game is known as Iro-Oni (艶鬼, Colored
  Demon) where players call out color and then slice the enemy where
  that color is displayed. However, if the color is not on the body of
  the person who called it out then the damage given is minimal; the
  more that the declared color is prevalent on the declarer's body, the
  more damage the attack will commit. A fifth game is called Daruma-san
  ga Koronda (だぁるまさん が こぁろんだ, The Dharma Doll Fell Down), a game
  similar to "Red Light Green Light". The sixth and final game is called
  Kageokuri (影送り, Silhouette Sending) which is a game where if a person
  is staring hard enough at another person's shadow, then their opponent
  creates after images of themselves. Katen Kyōkotsu's bankai is called
  Katen Kyōkotsu: Karamatsu Shinjū (花天狂骨枯松心中, "Heavenly Bloom Madness
  Bone: Withered Pine Lovers' Suicide"), which he uses only when his
  allies are at a safe distance from himself. When released,
  Shunsui's bankai covers a large area around Shunsui in an aura. This
  aura changes others' perception of the surrounding environment, making
  them perceive it as darkened, bleak and gloomy, and causes others to
  feel varying levels of melancholy and despair. While it maintains
  its shikai form, its bankai allows Shunsui to manifest various
  stories, which can kill opponent easily. It also manifests the spirit
  of Shunsui's zanpakutō, a Feudal-Japan-style mistress referred to as
  "Sakuranosuke" who created a silent kunoichi bodyguard from her being
  to hold Shinken Hakkyōken until Nanao eventually asks for it.




Second Division

Soi Fon (砕蜂（ソイフォン） Soi Fon, pinyin: Suì Fēng), born Fon Shaolin (蜂 梢綾（フォン シャオリン） Fon Shaorin, pinyin: Fēng Shaolíng)


  Soi Fon's Zanpakutō is Suzumebachi (雀蜂?, literally translated as
  "Hornet"). 
  
  Suzumebachi's Bankai, Jakuhō Raikōben (雀蜂雷公鞭, lit. "Hornet Whip of the
  Thunder Immortal") composes of a greatly enlarged version of the
  shikai form that encases the entirety of Soi Fon's arm, which she
  mentions as being bulky and incapable of concealing, and thus unsuited
  for use in missions requiring stealth. In this form, Suzumebachi
  becomes a long-range weapon and looks like a large rocket launcher;
  its main offensive capability allowing Soi Fon to launch a high-speed
  projectile that resembles a large missile. The thunderous recoil
  produced by the weapon requires Soi Fon to be anchored to a nearby
  stationary object via an enhanced cloth-like material (which doubles
  as armor) called Ginjōtan (銀条反, lit. "Stripe of Inverted Silver").




Third Division

Gin Ichimaru (市丸 ギン Ichimaru Gin)


  Gin's Bankai is "Kamishini no Yari" (神殺鎗 which means "God Kill Spear.")
  
  By adopting a stance where he firmly grips Kamishini no Yari with both
  hands and places the Zanpakutō's hilt squarely on the center of his
  chest, Gin is able to utilize the tremendous extension and contraction
  properties of his Bankai to an even more terrifying extent than its
  normal state. This occurs in the form of a piercing maneuver where the
  extension and contraction of the blade becomes almost completely
  unseen to even the most diligent of observers.


Shūsuke Amagai (天貝 繍助 Amagai Shūsuke)


  Amagai's Zanpakutō is Raika (雷火, roughly "Flash of Lightning").
  
  Raika's Bankai, Raika Gōen Kaku (雷火豪炎殻, literally "Lightning Flash
  Flame Shell"), enlarges the blade and creates a giant conch-shaped
  guard worn across Amagai's arm, with the handle of the blade hidden
  inside it. This form gives him greater control of his fire attacks,
  his most powerful technique creating giant dragons of fire to encircle
  and crush his foe.




Fourth Division

Yachiru "Retsu" Unohana (卯ノ花 烈 Unohana Retsu)


  Unohana's Zanpakutō is Minazuki (肉雫唼, roughly "purify the flesh").
  
  Her bankai is Minazuki (皆尽?, "All Things' End"). When activated it
  releases a thick, oily substance from the blade. Its powers have yet
  to be revealed. Minazuki appears as freed Zanpakutō spirit in the
  Zanpakutō Unknown Tales anime arc with Minazuki's form being a mute
  figure of unknown gender in a hooded robe, pretending it was
  imprisoning Yamamoto. Unohana has a scar on her collarbone inflicted
  by Zaraki Kenpachi when they fought prior joining the Gotei 13. She is
  also a skilled swordsman and have mastered all kind of swordsmanship
  techniques. In a new release of the manga, she is still shown fighting
  Zaraki Kenpachi. She then calls forth her Bankai, saying "Minazuki"
  and it appears that her blade is releasing blood as she slashes
  Captain Zaraki. It also appears that in an illusion, their flesh is
  peeling away from their bodies, and they are still fighting but as
  skeletons. It is still difficult and to early to truly understand what
  her Bankai actually does.




Sixth Division

Byakuya Kuchiki (朽木 白哉 Kuchiki Byakuya)


  His Bankai is Senbonzakura Kageyoshi (千本桜景厳, "Vibrant Display Of One
  Thousand Cherry Blossoms")
  
  When released, his blade disappears, and a thousand blades, resembling
  sakura petals, come upon his opponent and slash them multiple times.


Renji Abarai




  Hihiō Zabimaru turns his sword into a skeletal giant-snake like sword that
  has a head. It can dissemble and reform at Renji's will and can shoot
  focused reiatsu at the head.
  
  
  Sōō Zabimaru is compacted into form predomately worn by Renji
  with the snake skull smaller and affixed over Renji's right hand as a
  combination blade gauntlet and shield while a skeletal simian arm
  hovers just slightly above Renji's right arm. Sōō Zabimaru's signature
  technique Zaga Teppō is a close range attack where Renji uses his
  spiritual pressure to conjure a large snake skull that incinerates
  whatever it bites down on.


Source about Renji's bankai



Seventh Division

Sajin Komamura (狛村 左陣 Komamura Sajin)


  Komamura's Zanpakutō is Tengen (天譴, literally "divine punishment",
  translated in the English manga as "sword of heavenly retribution").
  
  Tengen's bankai, Kokujō Tengen Myō'ō (黒縄天譴明王, roughly "vidyarāja of
  black-roped divine punishment", translated in the English manga as
  "divine retribution, black ropes of ruination") brings forth the
  entire giant, which also copies Komamura's movements.




Eighth Division

Shunsui Kyōraku (京楽 春水 Kyōraku Shunsui)


  His bankai is unknown, but when asked by the Espada Coyote Starrk to
  show it to him, Ukitake warns him not to reveal it where so many
  people could witness it. Katen Kyōkotsu (花天狂骨) appears as a free Zanpakutō
  spirit in the Zanpakutō Unknown Tales anime arc as two separate female
  beings: a Feudal-Japan mistress and her silent ninja bodyguard,
  wielding Shunsui's swords.




Ninth Division

Kensei Muguruma (六車 拳西 Muguruma Kensei)


  Kensei's bankai is named Tekken Tachikaze (鐡拳・断地風, lit. Iron Fist
  Earth-Severing Wind). The combat knife transforms into two bladed


Kaname Tōsen (東仙 要 Tōsen Kaname)


  The Zanpakutō Tōsen uses originally belonged to his now-deceased
  friend and is called Suzumushi (清虫, literally "pure insects", but read
  as "cricket"; translated in the English manga as "bell bug").
  
  Suzumushi's bankai, named Suzumushi Tsuishiki: Enma Kōrogi (清虫終式紅閻魔蟋蟀,
  literally "Suzumushi, final style: Enma cricket") creates a large,
  black dome centered at Tōsen's location and anchored to the ground
  with ten rings. Anyone within the dome loses four of their five
  senses, retaining only the sensation of touch. The weakness of this
  technique is that whoever is touching Suzumushi's hilt is immune to
  the dome's effects, so anyone able to grab Suzumushi recovers their
  senses for as long as they remain in contact with the hilt.




Eleventh Division

Ikkaku Madarame (斑目 一角 Madarame Ikkaku)


  Ikkaku's Zanpakutō is Hōzukimaru (鬼灯丸, referring to the Japanese name
  for the winter cherry, meaning "demon light").
  
  Hōzukimaru's bankai, named Ryūmon Hōzukimaru (龍紋鬼灯丸, literally
  "dragon-crest demon light"), takes the form of three oversized weapons
  connected by a thick chain: two blades in each of his hands and a
  fan-shaped blade, adorned with a carving of a Chinese dragon, which
  floats behind him. Aside from the increase in power, his Bankai offers
  no special abilities, and is noted to be unusual for a bankai in this
  aspect.




Twelfth Division

Mayuri Kurotsuchi (涅 マユリ Kurotsuchi Mayuri)


  Mayuri Kurotsuchi's Zanpakutō is Ashisogi Jizō (疋殺地蔵, roughly
  "leg-cutting Buddha").
  
  Ashisogi Jizō's bankai, named Konjiki Ashisogi Jizō (金色疋殺地蔵,
  literally "golden leg-cutting Buddha", translated in the English manga
  as "divine leg-cutting Buddha"), takes the form of a giant caterpillar
  with a baby's head and a silver halo. The creature breathes a
  poisonous derivative of Mayuri's blood which is lethal to anyone who
  breathes it, aside from himself and Nemu.[45] It can also run through
  and skewer opponents with the multiple blades that erupt from its
  chest. Mayuri has physically modified his bankai so that it will
  self-destruct and return to its sealed state if it attacks him. It is
  also shown that Mayuri changes the poison he uses every time he shows
  his bankai.


Thirteenth Division

Rukia Kuchiki (朽木 ルキア Kuchiki Rukia)


  Hakka no Togame increases the area of influence of Rukia's
  ability to reach the temperature of absolute zero. Upon release, a
  pillar of cold mist rises up into the air from where Rukia is and
  covers a wide area in her vicinity. Anything within the area of
  influence of the mist is frozen. Once frozen solid, victims crumble
  away within seconds. Physical contact with Rukia herself causes other
  people to begin freezing solid because of how cold she is. While it is
  a powerful Bankai, it is also a difficult one, as the slightest
  mistake could kill Rukia. Because Rukia is physically affected along
  with everything else in her vicinity, the amount of time she can use
  her Bankai appears severely limited, as prolonged use can potentially
  cause her damage, so she is required to melt slowly when leaving this
  state, otherwise she would risk damaging herself.


Source about Rukia's bankai



General's Sources

Extras


Some images
As SingerOfTheFall pointed out in his answer about the one that appeared only in the anime.



  Shūsuke Amagai (天貝 繍助, Amagai Shūsuke)
  
  Raika Gōen Kaku (雷火 豪炎殻, Lightning Flash Flame Shell): Amagai holds
  out his Zanpakutō in front of him as fire swells around him. Raika
  becomes a large and more detailed version of its former self. Raika
  Gōen Kaku takes the form of an enlarged sword, and turns the edge of
  the blade a crimson red color, with a hook at the end. The inner side
  of the hook-blade sports several jagged edges. A large white shell
  with a light-purple trim covers the hilt. A massive hollow
  pauldron/shield with the handle is hidden behind it. The sword is
  about as tall as Amagai in this form.