Black coat

The Black Coat (sometimes called an Organization coat, or incorrectly as a black cloak) is a magical form of clothing that allows the person that wears it to safely pass through the Corridors of Darkness. It is most noticeable as the uniform for Organization XIII, but it is first used by Master Xehanort in Kingdom Hearts Birth by Sleep as an alternative to the Keyblade wielders' traditional armor. Several characters outside of the Organization wear this cloak, including Riku, King Mickey, and a Moogle, and it is unknown whether their coats were manifested, or simply taken from the Organization's spares.

The Black Coat's main purpose is to resist the corruption inflicted upon one's heart when they pass through a Corridor of Darkness. It also seems to be able to change shape to fit its wearer, as seen when Xion shifts from her normal form to Sora's.

Somehow, when Riku and King Mickey cast their cloaks off, the black gloves, pants, and boots worn under the cloak disappear. Furthermore, any trace of the wearer's clothing under the coat disappears, such as Riku's collar, jeans, and sneakers in his second attire, or Mickey's trademark white gloves. All these hint that donning the coat transforms whatever clothing the coat's bearer is wearing under it into the black gloves, pants, and boots seen on all the coat's wearers. However, Roxas's sneakers are visible under his cloak in several of his appearances.

Kingdom Hearts Birth by Sleep
Master Xehanort is seen wearing a black coat as he leaves Ventus, wrapped in white cloth, on Destiny Islands.

A new boss for the English version of the game wears a black coat.

Kingdom Hearts
The Unknown is seen wearing the black coat when he attacks Sora in Hollow Bastion.

Kingdom Hearts 358/2 Days
In Kingdom Hearts 358/2 Days, the Moogle wears the black coat, as a sign of it servicing Organization XIII.

Kingdom Hearts: Chain of Memories
Later, in Kingdom Hearts: Chain of Memories, Riku and King Mickey are given Black Coats by DiZ, so that they would not be tracked down by Organization XIII. Riku continues to wear the coat throughout Kingdom Hearts 358/2 Days and Kingdom Hearts II, until the explosion of the Kingdom Hearts Encoder returns him to his original form. What's interesting about Mickey's coat is that it is much smaller since it's catered to his body. Even the hood is sewn just for Mickey, with places for his ears.

Kingdom Hearts II
In Kingdom Hearts II, all the members of Organization XIII wear the black coat. King Mickey also dons the black coat until taking it off later in the game, and Riku is seen wearing it while being possessed by Xehanort's Heartless. He takes it off after reverting back to his original form. The Mushroom XIII are also seen wearing the coat.

Kingdom Hearts coded
In Kingdom Hearts coded, Jiminy's Journal takes on the form of Riku when the latter was wearing an Organization XIII coat.

Blank Points
When Aqua comes across Ansem the Wise at the Dark Margin in the Realm of Darkness, he is wearing the black coat.

Organization XIII
Among the most notable characters that sport the black coat are the members of Organization XIII. The Organization's coats are not uniform in appearance. For instance, Axel's coat has tight sleeves, while Xigbar and Demyx both wear coats with pointed shoulders. Additionally, Vexen's coat is longer than most other members, to the point that it covers his feet and its sleeves partially cover his hands. Roxas and Xion in Sora's form wear what appear to be black sneakers, while Larxene and Xion in her normal form wear heeled boots.

Trivia

 * The black coats were originally designed for Roxas and Riku in the secret video Another Side, Another Story... simply as a way to disguise them while they were still being designed for Kingdom Hearts II. It was again used simply to hide Xemnas's face during his optional boss battle in Kingdom Hearts Final Mix and in Deep Dive, starting the trend of all Organization members wearing it.
 * Nomura stated in an interview that his initial idea for "The Gathering" was to have Terra, Aqua and Ventus wear similar coats to the Organization XIII coats to hide their at-the-time undesigned faces, but felt that he could not use black coats once again, and so chose armor instead.