Carpenter
Duties probably include: Following architecural plans; rough and/or finish carpentry work; and minor plumbing and electrical work.
Lead Carpenter
Duties probably include: Rough and/or finish carpentry work; reading architectural plans and specs; supervising employees; solving problems on the jobsite; communicating with clients and subcontractors; and working with city inspectors.
Project Manager
The positions holds many of the duties of a Lead Carpenter, but may also: Create project schedule; verify plans for accuracy; work with architects to define details; coordinate subcontractors; confirm material lead times; resolve issue with clients; and handle punch list and job completion.
Production Manager
The position holds many of the duties of a Project Manager, but may also: Prepare cost estimate for each job; review contracts; conduct weekly meetings with Project Managers; evaluate subcontractors on an ongoing basis; oversee change orders; oversee tracking of actual costs against estimates; approve invoices for payment; maintain draw schedules; and participate in hiring decisions.