Most start-ups have a single individual who serves as leader; s/he could have any of a variety of titles; chief executive officer, president, and chief operating officer are typical. This individual reports to the board of directors. If you are considering having nobody in charge, or having two or more co-presidents in charge, you need to read Chapter 5 called “Role Dilemmas” in Wasserman [Wasserman, N., The Founder's Dilemmas: Anticipating and Avoiding the Pitfalls That Can Sink a Startup, Princeton, NJ: Princeton University Press, 2012].

Related Question:

What role should a founder play in a start-up?