Final answer:
The horizontal board referred to in the question, which encloses the ends of the rafter projections, is known as the fascia board. It provides a surface for gutter attachment and a finished look to the roof's edge.
Step-by-step explanation:
The horizontal board enclosing the ends of the rafter projections (or tails) is normally referred to as the fascia board. Both the soffit and cornice are also associated with the roofing and eaves structures, but the fascia is the correct answer in this context. The fascia is that long, straight board that runs along the lower edge of the roof, and it is mounted on the ends of the rafters, providing a surface for the attachment of gutters and it gives a finished look to the roof's edge.