How do I change an objects mesh without changing another objects mesh?

Whenever I attach a mesh to a GameObject, a large majority of my other objects change to replicate that mesh. This includes UDK meshes, and just about everything else. I figured the UDK items are linked somehow, but the new GameObjects I created change as well…

I’m 3 weeks new to Unity, so the solution might be easier than I think.

Instantiate the mesh, and change the instance.