Ring | Naruto | Akatsuki

Under Nagato [Uzumaki], each member of Akatsuki is given one of ten rings, to be worn on a specific finger. Other than apparently signifying one's membership, the rings seem to have some role in sealing the tailed beasts into the Demonic Statue of the Outer Path, indicating which of the statue's finger the members should stand on.

The rings are apparently important and irreplaceable: when Orochimaru defects he takes his ring with him, and for that reason he is never officially replaced; when Deidara loses the arm that his ring is on, he has greater interest in recovering the ring than the arm itself; Zetsu and Tobi specifically seek out Sasori's ring after his death, needing it for Tobi to replace him. When Tobi takes over, the rings are no longer important, as he simply discards it along with his damaged robe after battling Konan.

Despite all this, the rings' exact significance is never explained.
(from Narutopia)

The various rings and their wearing positions:

• Right thumb: "zero" (零, rei); worn by Pain. 
• Right index finger: "blue", "green" (青, ao, shō); worn by Deidara. 
• Right middle finger: "white" (白, byaku, haku); worn by Konan. 
• Right ring finger: "vermilion", "scarlet" (朱, shu); worn by Itachi Uchiha. 
• Right little finger: "sign of the boar" (亥, gai); worn by Zetsu.
• Left little finger: "sky", "void" (空, kū); worn by Orochimaru.
• Left ring finger: "south" (南, nan); worn by Jūzō Biwa^{(anime only)} and later by Kisame Hoshigaki. 
• Left middle finger: "north" (北, hoku); worn by Kakuzu. 
• Left index finger: "three" (三, san); worn by Hidan. 
• Left thumb: "jewel", "ball", also the black king in shōgi (玉, gyoku); worn by Sasori and later by Tobi.