Select Contained within will select all objects in the target set that are entirely contained within an object in the source / argument box set. Select Containing will select all objects in the target set that completely contain any object in the source / argument box set.
Suppose we have a saved selection called Durango that selects the province of Durango in Mexico.
Select Contained within
Running Select Contained within selects the two points and the line that are entirely within Durango.
Suppose we have another layer in the map containing circular areas. Running Select Containing selects the large circular area that entirely contains Durango.
Select Adjacent to