“Why ... why did you put on that ring? It carries a curse, surely you realized that. Why even touch it? ...There is no halting such a spell forever. It will spread eventually, it is the sort of curse that strengthens over time."
-- Snape to Dumbledore (DH33)
The curse on the ring refers to a deadly curse placed by Voldemort on Marvolo Gaunt‘s ring to protect the Horcrux within. The curse was lethal and would have killed Dumbledore almost immediately after he put the ring on if Snape hadn’t performed a counter-curse to slow it’s spread.
Tom Riddle used the ring of his grandfather, Marvolo Gaunt, to make a Horcrux to encase part of his torn soul. The stone in the ring was the Resurrection Stone, one of the Deathly Hallows passed down from Cadmus Peverell (HBP10, TTB). The temptation of being able to use the Stone was too much for Dumbledore to resist. He put on the ring because he thought the stone would give him a chance to see his dead mother and sister again (DH33,DH35). However, the Stone was useless while trapped inside the cursed Horcrux ring because it had to be turned three times in someone’s hand in order to work (BLC).
Instead, when Dumbledore put on the ring, it made his hand wither and blacken, and Snape had to work quickly to keep Dumbledore from dying immediately (HBP23). According to Snape, the curse on the ring was powerful and could not be stopped forever, but he managed to give Dumbledore one year to live (DH33).