Need the names of materials or any suggestions as to what I should do.
Underneath the drip-edge is the Fascia board. This is usually 1 x 4 inch stock (looking at your photo).Fascia board can be of any variety of wood or composite although Pine can be painted to protect it from the elements, Redwood and other weather resistant wood species can be used to increase longevity.
Below the Fascia is a Trim molding. This is sometimes used to close large gaps between framing and/or finish pieces or to add more detail to the structure.
Under the Trim is the siding (aluminum or vinyl). The vertical trim piece is missing. It is installed to cover the ends of the horizontal siding and neaten their appearance.
Beside the siding the doors weatherstripping is thrashed and should be repaired or replaced.