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.