What I did was just that I created a new product and then I asked catia to insert different "stuff" thanks to the insert tab. And I saw that we had a large choice of insertions and I wanted to know the difference between these many choices.
There is no design or other approach at this time. I would say that it is more a process of knowledge of the basics of the software.
It does not contain geometry (it is a link pointer)
It contains the assembly constraints of the parts, S/E and component.
It can contain user settings.
It can contain scenes (exploded example).
The "Composent" Screw Grouping (Example)
It behaves like the CATProduct except that we don't have a file.
One can copy a component from one CATProduct to another (with or without an associativity link)
The CATPart is the part file.
But not only that, you can very easily design for example a multi-material part, or for example a cylinder and have the constraints as if you were in an assembly.
(It's less user-friendly than in the ASSembly workshop, but when you don't want to manage as many parts files + assembly it's an effective solution).
Just to complete franck.ceroux's answer on the icons, the first icon "Piece 1.1" is "the instance". It's a number that increments every time you insert the same part in a product or component. The second icon is the definition file of the part itself.
If you switch to the visualization mode, only the first icon will appear.
In design mode, if you want to edit your part from the product, you have to double-click on the second icon.