Select the resource you want to update to the latest version, then select Update at the top of the Resources table. This will update all versions of the selected resource that are used or cited below the selected table of contents item to the latest version of the resource. So, if you selected a Lesson, selecting Update would only update to the latest version of the selected resource in that lesson and any sections within that lesson, but not anywhere else that resource is cited within the initiative.
Note: You can only update to the latest version of the resource that is in LEAD. So, if you have three versions of a resource, versions 1, 2, and 3, and version 1 is currently cited/used in your initiative, you can ONLY update to version 3, you cannot update to version 2.
Once the update process has finished, the latest version of the resource is displayed in the list (now without a flag) and any previous versions of the resource are no longer listed (though those may still be listed, if you select a different table of contents item). The New Version flags
that show you where the resource has been used or cited in the selected table of contents item change to System flags 
to show you that an object cited by this item has been changed.
After you review the item that uses or cites the resource, you can either delete the flag on the Explain Flag screen if no change is necessary, or edit the item that uses or cites the resource. If you are on version 2 or higher of the initiative, you will get a User Edited flag 
after saving the edits to the item.