The "mouth" of a ko is the place where the repeated captures occur. The term is very rarely used, and then only in rules and their explanations.
The basic ko rule only forbids recapture of a single stone, so a ko mouth will always consist of two adjacent points. These are all the possible shapes for creating a ko mouth on a conventional board, created by shifting the basic position (a) to an edge or corner; general graph go may open other possibilities.
Sometimes a ko mouth can be adjacent to another. In the pictured case white can capture in the ko marked with squares only after playing in the circled ko. This kind of situation is called a multi-stage ko.