In the book Unwind, why is Connor an important character?
Connor is an important character, because he is one of the three main protagonists. The other two main protagonists are Lev and Risa. Most of the chapters in the book are focused on one of those three characters, so Connor is important simply because he has a lot of page time.
In terms of plot, Connor is important because his actions propel the story forward. He might not see himself as a leader for most of the novel, but he can't help but drive events and force characters to adapt to his actions. For example, Connor forces Lev to begin evaluating the unwind process and the parental practice of tithing their children. Connor does this by "kidnapping" Lev. Connor forces Risa into similar difficult circumstances when he "adopts" the storked baby.
By the end of the novel, Connor has become the leader of the Unwinds that support the Admiral. That puts him in direct conflict with Roland, and drives the book toward its ultimate climax. At the end of the novel, Connor returns to the Graveyard and becomes the new Admiral, which effectively sets up a sequel novel.