The majority of drop down menu options are controlled under Customization > Drop Down Menus. Group names have a dedicated section under Customization > Groups

In the Admin section, when you click on Groups, you will see the current list of group name options.

Adding a group name

Type you a new group name in the text box and click Add

Adding constituents to a group

There are two ways you can add constituents to a group name. The first way is to go individual into constituent records, navigated to the groups page, and select the group name. The second way is to add constituents en bulk to a group name - this is done by clicking the add to group link.

Clicking the Add to Groups link will present you with the Search page, where you can select your criteria. You'll be presented with a list of matching constituents, where you simply click Add. All those constituents will then be flagged into this group name.