Half-Moon Spectacles – Harry Potter Lexicon

"He had several feet of long silver hair and beard, half-moon spectacles, and an extremely crooked nose."
-- description of Albus Dumbledore (CS5)

Dumbledore wears half-moon shaped glasses.

Generally this type of eyewear is used for reading, like the bottom-lens of a pair of bifocals. But Dumbledore wears his half-moon spectacles at all times, even at night. For instance, on the evening when Hagrid brought baby Harry to Privet Drive, he was wearing them:

His blue eyes were light, bright, and sparkling behind half-moon spectacles and his nose was very long and crooked, as though it had been broken at least twice. This man's name was Albus Dumbledore (PS1).

Dumbledore also uses his spectacles to "study" or "survey" things and people.

The tip of Dumbledore's long, crooked nose was barely an inch from Mrs. Norris's fur. He was looking at her closely through his half-moon spectacles, his long fingers gently prodding and poking (CS9).

"I suppose he's told you the same fairy tale he's planted in Potter's mind?" spat Snape. "Something about a rat, and Pettigrew being alive --"
"That, indeed, is Black's story," said Dumbledore, surveying Snape closely through his half-moon spectacles (PA21).

Yet other times Dumbledore let the half-moon specs slide down his nose so he could look "over" them, often when he was attempting Legilimency. For instance when he wondered how many of his memories Harry had seen in the Pensieve:

Harry saw his own face change smoothly into Snape's, who opened his mouth and spoke to the ceiling, his voice echoing slightly.
"It's coming back . . . Karkaroff's too . . . stronger and clearer than ever..."
"A connection I could have made without assistance," Dumbledore sighed, "but never mind." He peered over the top of his half-moon spectacles at Harry, who was gaping at Snape's face, which was continuing to swirl around the bowl.

Or when he wondered if Harry was trying hard enough to get the missing memory about the Horcruxes from Professor Slughorn:

"Well, I asked Professor Slughorn about it at the end of Potions, sir, but, er, he wouldn't give it to me." There was a little silence.
"I see," said Dumbledore eventually, peering at Harry over the top of his half-moon spectacles and giving Harry the usual sensation that he was being X-rayed. And you feel that you have exerted your very best efforts in this matter, do you? That you have exercised all of your considerable ingenuity? That you have left no depth of cunning unplumbed in your quest to retrieve the memory?"
"Well," Harry stalled, at a loss for what to say next (HBP20).

Here's an instance when Umbridge thought she had Dumbledore cornered in his office, and perhaps he was trying to find out what she knew of Dumbledore's Army:

'I think you'll find you're wrong there, Dolores,' said Dumbledore quietly, peering at her over the half-moon spectacles perched halfway down his crooked nose (OP27).