There is no real consensus on the role of a COO in startups. His or her responsibilities change from company to company.
I myself held this position for about 15 years in the two startups I co-founded. The main characteristic I give it is its catch-all scope.
A bit like the last rule of a firewall or a web server configuration that processes the remaining cases after all the preceding rules have been applied. The COO takes care of everything that is not taken care of by someone else. Then he or she delegates portions of his or her responsibilities as managers are appointed to handle them.
At least that's how I've experienced it, and the conversations I've had with other co-founders who have worked in this position seem to confirm this description.
I've selected a few interesting articles for you to get acquainted with, or learn more about, this role.
This HBR article from 2006 provides an excellent review of what makes a COO essential to a company. It's not written with startups in mind, but is still just as relevant.
The answers in this Quora thread provide interesting insights into the different responsibilities that can fall to a COO.
In this post, Andy Sparks explores the ambiguity of the COO role, a position he discovered and held at Mattermark. He describes how his role evolved with the needs of the company to become its COO.
Mark Suster considers a COO to be overkill for a startup. He believes that in this context the CEO should handle most of the tasks that are assigned to the COO in large organizations. In this quality post, he uses his own experience to support this opinion.