Turbine inlet nozzle guide vanes are internally cooled and film cooled using high pressure compressor bleed air. The bleed air enters an internal passages at the end of the guide vane. Some internal passages have an S-shape and makes several passes from one end of the vane to the other to maximize heat transfer to the cooling air. Some airflow is directed thru holes to provide intensive impingement cooling of internal faces where exterior surface temperatures are highest. The majority of the internal airflow is discharged thru a series of small holes to provide film cooling of the guide vane or blade.
I think the term "film cooling" is slightly misleading since it does fully describe how the approach works. While the boundary flow provided by the air discharged from the holes does indeed result in some cooling of the adjacent metal surface, the boundary airflow also acts as a very effective form of thermal barrier that minimizes heat transfer from the hot core gas flow to the vane surface.
Here is a schematic the shows the cooling airflow to the turbine inlet nozzle guide vanes and HP turbine blades.
Below is an image of the internal airflow in a HP turbine blade:
