This topic should not be confused with the Group feature which allows you to arrange/combine multiple elements into a repeateable/reusable/easily updated Group.
When you make Revit content you inevitably have to make a parameter of your own. Naming parameters is a can of worms and I'm leaving that lid on for this post. I'm going to open the lid of a smaller can but one that doesn't get much attention, because of its size I suppose. I'm referring to the highlighted feature in the image. The Group parameter under drop down list box.
Once you've made a choice you see the results in the Properties Dialog like this one, created My Parameter using Group: Title Text.
The blue bars are a Group header/parent and the parameters listed beneath it are its children. Clicking on the bar will collapse or expand the list of parameters assigned to the group. Groups only show up if there are corresponding parameters
So how do you choose or decide? It is an organization tool, not a functional feature that causes something to happen (other than sorting the parameters themselves). If it helps, you are defining what the parameter's role is or said another way, what is its purpose?
If you are applying parameters to change the size or dimensions of a family then the logical group (to me) or role is dimensions. A constraint is a control role...this parameter is limiting or managing behavior elsewhere. If you are creating a Yes/No parameter to control the visibility of a solid that you could choose the Graphics group since it alters the graphic representation of the family. The choice is subjective...which means you (your firm, your clients) need to reach some agreement about how to apply them.
A lesser aspect, but the Group value also defines where the parameter appears in the list of all the parameters because of how Revit sorts the list. I know some users who put things in Constraints just because it is FIRST and they put things that they don't want users to find (unless they actually scroll down the list) in Other because it is LAST. Just to make it a bit more confusing...the sorting of the Groups is not purely alphabetical, they are displayed in what I assume is a hardwired order.
This feature is one of the few choices you are faced with while creating a parameter that you can make and then change afterward. Happy choosing!