NFL: Lost - Need an Answer From NFL Experts and Gurus Here

They're like company managers, I suspect.  They may not do the actual work, but they tell the coordinaters what they want.

Also, when problems arise with the offense, defense, special teams, etc, it's ultimately their responsibility to resolve it.

During the week, the head coach may even help design the offensive and/or defensive plays, then let the coordinators choose which plays to run during the actual game.