Let's begin with the most obvious part: his talent. He is, to put it simply, an incredibly gifted actor.
He's a perfect blend of character actor and leading man. That's a very rare breed. He's versatile to a chameleon-like degree (watch Dummy and Predators back to back and you'll see exactly what I mean). He completely immerses himself in his work. He becomes his characters and loses himself in that moment. Yet he retains the charisma and recognizability of a "movie star," as well. Gifted actors and movie stars are rarely the same people these days. Adrien Brody is both.
He draws you in, then makes you care deeply about what's happening on screen. Because he cares. You can feel it. The emotions and situations become very real for the audience. That is his gift.
Adrien's caring and empathy extend into his off-screen life, as well. He lends his time to many causes, such as: AIDS research, stopping the use of carriage horses in New York City, and ending damaging palm oil harvesting in Sumatra. You can't help but admire someone who uses their celebrity to draw attention to issues, instead of just attention for themselves.
Then there's the persona. I would say personality, but since I don't know him personally, I'll go with the word persona (which is to say: his public personality). He exudes both confidence and humility in equal parts. He's a sensitive romantic and a "bad boy." An intelligent deep thinker, who probably also appreciates a good dirty joke now and then. He has both a shyness and cockiness to him.
His eyes convey both sadness and joy in the same expression. I'm not sure how he pulls off this type of balance, but it's very unique and admirable. And captivating.
That is just a part of why I am so intrigued by this man. Feel free to visit the various pages here to find out more about the man and my fandom.