According to the classical theory of concepts, concepts have are complex mental representations that have a definitional structure (the concept fits properties according to its definition).
Other theories developed such as the prototype theory and the dual theory. The prototype theory suggests that concepts fit a stereotypical prototype (kind of like Jungian archetypes). The dual theory developed later and suggests that concepts have a 'core' and an 'identification procedure'. However 'Conceptual Atomism' suggests that concepts don't have any structure and they just correspond with what they represent - be that casual or historical.
In another article of mine m52492 I reference James Sully (1892) who points out that concepts have three parts - abstraction, comparison and generalization. That suggests that people go through a process of thinking for each concept - they compare the concept to other concepts, generalize the concept - but first just have an abstract picture of the concept.
His simple explanation basically just points out that concepts involve thinking 'more' about each concept - including generalizing it. The other theories of concepts all have valid points as well however. The prototype theory suggests generalization is involved (as prototypes are generalized). And the dual theory suggests that identification is involved (which is similar to comparison). While the atomism theory suggests that it is just a simple thought process - which would be the 'abstraction' phase of understanding the concepts.