I'm fairly new to Blender. Over to the right of the screen (in the outliner) I have a couple of 'layers' I think they are, and I really need to combine them into one. I want to go from this:
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If HalfKiloByte is correct and you want to collapse the levels of the tree view, you do that by pressing the icon to the left of the name, at which point the hyphen in the circle will be replaced by a plus sign in a circle icon. For a more thorough explanation of the outliner editor window, you might want to view Neil Hirsig's tutorial.
However, the items to which you refer in the two illustrations are different. In the upper illustration accompanying the original post, the upper triangle with an orange background, whose name is (in part) "Mesh 29 Group 30 Grou" is the name of an object. The next line down, whose name is (in part) "Mesh 29 Group 30 G" is a mesh, which at the moment is the only constituent element of the object named in the line above. Both meshes and objects are namable, but they cannot be merged, as they are different date types. The lower illustration is of a different mesh and object, which appear to have the same name "luger". The only differences between the second illustration and the first is that the first illustration exhibits an object containing a mesh which has (at least) five materials assigned to it, and possibly more materials, or other data elements—textures or modifiers, while in the second illustration, there are no other data elements assigned to the mesh (evidenced by the lack of an icon containing a plus sign in a circle), and that the mesh object is opened in edit mode (evidenced by the fact that the triangle next to the second occurrence of the name "luger" is on a white background.
If you want the object and mesh in the first illustration to look like the second, delete all subsidiary material elements.