The injury to his hand is done by a powerful curse on the ring horcrux. The ring had been Tom Marvolo Riddle's grandfather, Marvolo Gaunt's, ring which he had inherited through centuries from Cadmus Peverell. Dumbledore recognised the Resurection Stone set in it and was blinded momentarily by the longing to see his parents and sister to apoligize to them. Severus Snape stopped the spreading of the curse temporarily at the wrist. But said that it would grow stronger with time, spread and kill Dumbledore within a year.