The interior of a geometry is all points that are part of the geometry except the boundary.
The exterior of a geometry is all points that are not part of the geometry. This can include the space inside an interior ring, for example in the case of a polygon with a hole. Similarly, the space both inside and outside a linestring ring is considered the exterior.
The boundary of a geometry is what is returned by the ST_Boundary method.
Knowing the boundary of a geometry helps when comparing to another geometry to determine how the two geometries are related. However, while all geometries have an interior and an exterior, not all geometries have a boundary, nor are their boundaries always intuitive.
Here are some cases of geometries where the boundary may not be intuitive: