Press the plus button to create whatever kind of group makes the most sense for whatever you have selected. For instance, while in Object Mode this creates a collection of all selected objects. Pressing it while in Mesh Edit Mode creates and assigns a vertex group from the selected mesh. Pressing it while in GPencil Mode creates a layer from the selected points. Pressing it while in Pose or Armature Edit Modes creates bone groups from selected bones. Hold Shift before clicking the plus sign to name the new group before creating it.
If an object possess at least one vertex group, the enable/disable vertex group selection button will appear next to the plus. This functionality helps keep the menu layout clean as well as provide an indicator that the object you are editing possess a group. This works for other group types such as bone groups as well. While this button is active, double-clicking on any member of the group will cause it to become selected. If disabled double-click group selection within the object will be ignored.