In the 5th book/movie, Harry and other students made a secret organization. They signed their names on a paper titled "Dumbledore's Army".
When the paper was discovered by Inquisitorial Squad and Umbridge, Dumbledore lied to Minister for Magic to save students. He said that he instructed Harry to build this army and the paper clearly says Dumbledore's Army, not Harry's Army.
But, did Dumbledore know about this secret organization prior to this event?
Remember that Albus's brother, Aberforth, owned and ran The Hog's Head, where the first Dumbledore's Army meeting was held. Aberforth Dumbledore was in the room when the meeting was held.
The barman sidled towards them out of a back room. He was a grumpy-looking old man with a great deal of long grey hair and beard. He was tall and thin and looked vaguely familiar to Harry.
(Order of the Phoenix, Chapter 16, In the Hog's Head).
We know that Aberforth was in the habit of slipping his brother information about his customers.
"Then if I were to go to the Hog's Head tonight, I would not find a group of them - Nott, Rosier, Mulciber, Dolohov - awaiting your return? Devoted friends indeed, to travel this far with you on a snowy night, merely to wish you luck as you attempted to secure a teaching post."
There could be no doubt that Dumbledore's detailed knowledge of those with whom he was travelling was even less welcome to Voldemort; however, he rallied almost at once.
"You are omniscient as ever, Dumbledore."
"Oh, no, merely friendly with the local barman," said Dumbledore lightly.
(Half-Blood Prince, Chapter 20, Lord Voldemort's Request).
As Anthony Grist says, Mundungus Fletcher was an Order member who was also at the same meeting so Dumbledore could have heard it from him. Or from both Mundungus and Aberforth.
We know that when he was confronted with the group's existence by Fudge that he was very open about the fact that Harry held a DADA meeting in The Hog's Head.
"Cornelius, I do not deny - and nor, I am sure, does Harry - that he was in the Hog's Head that day, nor that he was trying to recruit students to a Defence Against the Dark Arts group. I am merely pointing out that Dolores is quite wrong to suggest that such a group was, at that time, illegal."
(Order of the Phoenix, Chapter 27, The Centaur and the Sneak).
I think it's doubtful that Dumbledore would've been so forceful in making that statement if he had been completely oblivious of the group's existence before that point.
We can't be sure whether or not Dumbledore was aware of the DA meetings that took place in Hogwarts. But we do know that Dumbledore was aware of most things that go on at Hogwarts and that he was keeping a close eye on Harry, so he probably knew about those meetings too.
We know that Mundungus Fletcher overheard the original plans for a secret Defence Against the Dark Arts group in the Hog's Head, and that he then told Sirius.
"Not that good," said Harry, as Hermione pulled Crookshanks back to stop him singeing his whiskers. "The Ministry's forced through another decree, which means we're not allowed to have Quidditch teams —"
"— or secret Defense Against the Dark Arts groups?" said Sirius.
There was a short pause.
"How did you know about that?" Harry demanded.
"You want to choose your meeting places more carefully," said Sirius, grinning still more broadly. "The Hog's Head, I ask you..."
"Well, it was better than the Three Broomsticks!" said Hermione defensively. "That's always packed with people —"
"— which means you'd have been harder to overhear," said Sirius. "You've got a lot to learn, Hermione."
"Who overheard us?" Harry demanded.
"Mundungus, of course," said Sirius, and when they all looked puzzled he laughed. "He was the witch under the veil."
Harry Potter and the Order of the Phoenix, Chapter Seventeen
It's possible that Mundungus only ever told Sirius, who didn't share that information with anybody else, but this seems unlikely. For a start, Mundungus doesn't seem particularly good at keeping secrets, and Sirius is likely to share this information with Dumbledore, since it would likely be relevant information in order to keep Harry safe.
