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.
How can a title accommodate such diversity and still be meaningful? Answering that question requires a shift in perspective. The key is in the orientation of the role. While other jobs are primarily defined in relation to the work to be done and the structure of the organization, the COO’s role is defined in relation to the CEO as an individual.