How did Dumbledore injure his hand? Why did he ask Snape to contain the curse in his hand, instead of asking Mad Eye Moody to do so? After all, Mad Eye is a close friend of Dumbledore. Why did Dumbledore choose to ask Severus Snape to deal with the curse?
Dumbledore injured his hand by putting on the cursed ring of Marvolo Gaunt, which contained a curse guarding the Horcrux.
He asked Snape to help him with the curse, for the following reasons:
Severus Snape is a Potions master, and a very good one, while Mad Eye is more of a fighter, detective, whatever, but not as good with potions and healing. Perhaps, Snape was the only person Dumbledore could think of, that was able to save him. The school matron was only second to him.
Dumbledore trusted Snape, as it is revealed in Book 7, when Harry sees Snape's memories in the Pensieve. Snape's grief on Lily's death was quite sincere, giving Dumbledore reasons to trust him completely.
Nobody should be aware of Dumbledore's research on Horcruxes, except for his trusted allies (including the Trio). If, say, Dumbledore went to St.Mungo's with his injured hand, it could leak critical information to Death Eaters, both on his research and imminent death. It was very important that this information was kept secret, and Snape was the best man you could think of to keep things secret. His capabilities in Occlumency were exceptional, he was not compromised until the end.