The Analysis and Geoprocessing Tool Gallery - Featured group is a subset of the items found in the Analysis and Geoprocessing Tool Gallery group. The featured gallery is what appears on the Analysis and Geoprocessing Resource Center as illustrated below.
Items in the featured gallery are those that we (ArcGISTeamAnalysis) find most interesting and useful. On occasion, we scan the Analysis and Geoprocessing Tool Gallery group for those items that are useful, of high quality, or are highly rated with good comments. We then share the item to the featured gallery.
The featured gallery is a private group—you cannot request membership like you can with the Analysis and Geoprocessing Tool Gallery.