How do I make this damask pattern on the cake? To me it appears to me fondant. See the attached photo.

See this stencil:
http://www.globalsugarart.com/product.php?id=23052&name=Damask%20Side%20Stencil%20by%20Designer%20Stencils

I think you squeegee something soft, like buttercream, over the stencil. I might be possible to thin fondant down to where you could easily use it with the stencil.

OK, that is just classic royal icing stenciling with the pattern she linked you to. Maybe black buttercream...but it's one of the two.