Tough question. It's been debated quite a bit too, so it's okay to disagree with this answer. I do not believe that Big Brother is an actual who. I think he is the symbolic image of the leader of Oceana. Big Brother is believed to be the ruling head of the government, but I don't think he really exists. His posters and face just serve as the physical embodiment of the oppressive government.
"It depicted simply an enormous face, more than a metre wide: the face of a man of about forty-five, with a heavy black moustache and ruggedly handsome features."
He could be a real guy. I'm not completely ruling it out, but I've always found it odd that the posters never changed. It's always the same image.
"It was one of those pictures which are so contrived that the eyes follow you about when you move. BIG BROTHER IS WATCHING YOU, the caption beneath it ran."
The above quote also makes me think Big Brother is symbolic. I'm somewhat of a skeptic anyway, so when I read the line "Big Brother is Watching You" I've always thought "it's not possible for one guy to watch everybody." But it is possible for an entire oppressive government regime with a large police force and surveillance everywhere to watch you. That's what Winston is dealing with throughout the story. There are very few places that he can go without a telescreen watching. In that regard, thinking of Big Brother as an institution really does make it possible for "Big Brother is Watching You" to be true.
"The telescreen received and transmitted simultaneously. Any sound that Winston made, above the level of a very low whisper, would be picked up by it, moreover, so long as he remained within the field of vision which the metal plaque commanded, he could be seen as well as heard. There was of course no way of knowing whether you were being watched at any given moment. How often, or on what system, the Thought Police plugged in on any individual wire was guesswork. It was even conceivable that they watched everybody all the time. But at any rate they could plug in your wire whenever they wanted to. You had to live — did live, from habit that became instinct — in the assumption that every sound you made was overheard, and, except in darkness, every movement scrutinized."