Although I knew the general plot through my outline, around midway through the writing of my book, my characters started doing things that I had not planned for them. They surprised me by certain decisions and or actions. This was when I knew that I had created dimensional characters that came off the page, that came to life.
For example, I did not know that Cosmo would do a magic trick for Isabella in the final chapter or that Silvia would decide to become an art school teacher, or that Donna would change her mind about the party. I did not know that Frank would be the first to show up at the party and did not know that Donna would decide, definitely, to leave Frank. When I was ten years old, I wrote a story entitled The Magic Pen. In the story, the main character had a pen, that when used, would create stories as if by magic.
I actually found myself hanging out with my characters in a way. I would be walking down the street and imagine that I was in a dialogue with them. Or I would talk about them to myself like they were real people. A lot of my thoughts went into working out how they would act or what they would say. And eventually they became so real and so dimensional that they started doing things on their own! Kind of creepy, but super cool.
See great article on creating characters in this Guardian article: http://www.guardian.co.uk/books/2011/oct/16/how-to-write-fiction-andrew-miller