This is a huge topic, so I'm going to boil it down to what I feel are the basics that make up compelling characters.
I think the most compelling characters are the ones who are the most developed, the "roundest," as they often say. But those terms don't really do the concept justice.
They are not easily figured out by what they look like on the outside. They think, feel, and have depth, layers. (Like those from The Help.)
What others think of them isn't necessarily the entire truth. (How about that Mr. Darcy?)
Their background and past experiences have shaped and defined them, and those things impact the current story and how they behave in it. (Easy example: Harry Potter.)
A compelling character isn't perfect, but is doing their best, even if that means standing facing danger, public ridicule, or something else. (Montag sure has a fight.)
It's someone I can relate to, even in a small way, and even if they're in a bizarre situation I'll never be in. (Katniss, anyone?)
A compelling character needs weaknesses and flaws, but ones that aren't so huge I dislike the person. To balance the flaws, they need a redeeming quality. (We'd hate John Cleaver without his redeeming quality. With it, he intrigues us.)
I like seeing characters who have weaknesses that can be turned on their heads to become strengths in the final, crucial moments of the story. (Stealing from myself here: this one is very Tabitha.)
Most of all, a compelling character is someone who has become so real to me that even after I read the last page and close the covers of the book (or turn off the Kindle, as the case may be), I keep thinking about them for a long, long time.