Like many things, ship surveying can be defined in a few different ways. I can only speak for myself, so here it goes.
I started as a Marine Surveyor with the American Bureau of Shipping, which is an independent Class Society. As a surveyor with the Bureau, vessels are examined during construction to verify compliance with approved plans. Vessels are also examined after construction. These exams fall into different categories. Annual Surveys, Dry Docking Surveys (every 2 1/2 years) and Special Surveys (every 5 years). Vital machinery components are also examined every 2 1/2 or 5 years and subject to separate surveys, such as tailshafts and boilers. Class Society surveyors are also authorized to act on behalf of the flag states that register the vessels to carry out inspections and issue trading certificates for the various SOLAS and MARPOL conventions (safety and polluton prevention) as well as Load Line conventions.
Class Surveyors also carry out damage surveys subsequent to incidents in order to determine the repairs necessary to return the vessel to a satisfactory Class condition.
Independent Surveyors can report to either Vessel Owners, Charterers or Underwriters. An independent surveyor will carry out damage surveys much like a Class Surveyor, however the extent of the repairs recommended would then be to return the Vessel to the condition prior to the incident, and may often include items not included in a Class Survey. Indepenent surveyors also carry out on-hire (or charter) and off hire (or charter) surveys to determine if there was any damage to a vessel while it was in the hands of someone other than the owner. There are also condition surveys, pre-purchase surveys, valuation surveys, etc.
Marine surveyors can also carry out cargo surveys relating to the condition, stowage and discharge if various break bulk, bulk and liquid cargoes.
Most of the Class Society Surveyors are engineers. Some hull surveyors are from the deck department and most cargo surveyors are from the deck department.
I am leaving out quite a bit, but that is a start.