Character Analysis
Dumbledore is everything a bright young wizard could need: smart, nurturing, calming, strict (when need be), and protective of his students. However, even he is having a hard time dealing with the, er, special challenges that the Triwizard Tournament has thrown into the school year—challenges that set the stage for even more problems that he'll have to solve in the future.
"He's A Genius"
At the beginning of the year, Dumbledore is dealing with normal headmaster stuff. Well, normal for wizards. The school is hosting other students and headmasters for the Triwizard Tournament, and some of the students are disgruntled because they don't meet the age requirement (seventeen) for participation.
When the Weasley twins decide they're going to try to get around the (magically imposed) restrictions on entering their names for consideration to compete for Hogwarts, they discover that Dumbledore is definitely cleverer than they are.
Hermione, another smarty, is quick to warn them that they aren't bright enough to outwit Dumbledore:
HERMIONE: It's not going to work.
GEORGE: Oh, yeah? And why is that, Granger?
HERMIONE: You see this? This is an Age Line. Dumbledore drew it himself.
GEORGE: So?
HERMIONE: So, a genius like Dumbledore couldn't possibly be fooled by a dodge as pathetically dimwitted as an aging potion.
You can probably see where this is going. The spell backfires and makes them look like old men (think Statler and Waldorf from the Muppet Show).
It's a funny trick, but it's also a good reminder to all the students: Dumbledore is definitely a brainy one, even if he doesn't (always) feel the need to show off.
Wizard Troubles
It's definitely not all fun and games at Hogwarts for Dumbledore, though. In fact, it's not even mostly fun and games, what with all kinds of ominous things happening.
Dumbledore apparently has a trick for preventing himself from getting overwhelmed by all the worries and potential problems that surround him. He uses something called a Pensieve to stash thoughts and memories. It kind of makes room in his head so the old wizardly noggin doesn't get overloaded.
He catches Harry taking a peek into the Pensieve and explains how it helps him:
DUMBLEDORE: Curiosity is not a sin, Harry. But you should exercise caution. It's a Pensieve. Very useful if, like me, you find your mind a wee bit stretched. It allows me to see once more things I've already seen. You see, Harry, I have searched and searched for something—some small detail, something I might have overlooked, something that would explain why these terrible things have happened. Every time I get close to an answer, it slips away. It's maddening.
Oh right, we forgot to mention: in addition to storage, Dumbledore actually uses the Pensieve to revisit and sort through thoughts to make some sense out of them. Sounds pretty handy (and cheaper than therapy, which is how non-wizarding adults achieve the same effects).
He's Not Afraid to Battle The Ministry
Dumbledore isn't one to tow the party line, generally speaking—which is why Professor McGonagall is so shocked when Dumbledore refuses to stand up to Barty Crouch Sr. about letting Harry Potter compete in the Triwizard Tournament:
DUMBLEDORE: What do you suggest, Minerva?
MCGONAGALL: Put an end to it. Don't let Potter compete.
DUMBLEDORE: You heard Barty; the rules are clear.
MCGONAGALL: Well, the devil with Barty, and his rules—and since when did you accommodate the Ministry?
It seems that Dumbledore likes to play nice if he can…but that rep as someone who doesn't "accommodate" those in authority just 'cause will probably be important as the danger Voldemort poses grows.
In any case, we're super glad that Harry has someone like Dumbledore in his corner.