Stringers provide support to a staircase and there are many different varieties (see below for the different varieties).
Stringer fascia is the finished surface that on a western style stringer is on the inside vertical surface of the stringer.
A stringer capping is the finished surface that is on the top face of a western style stringer.
Shoe mould is moulding that covers the gap between the shoe and the drywall surface of the wall below where the shoe is installed.