Coarse geometry is...
...what you see when you screw up your eyes. Take an impressionist painting. Up close it doen't make any sense, but from far away it seems to have lots of detail. Now move the painting further and further away, off towards infinity. If the image is infinitely large, then it will still have structure and geometry in the limit; this is the coarse geometry of the image.
N.J. Wright, My thesis.
N.J. Wright, C0 coarse geometry and scalar curvature. Journal of Functional Analysis, 197 (2003) 469–488 doi:10.1016/S0022-1236(02)00025-3.
N.J. Wright, The coarse Baum-Connes conjecture via C0 coarse geometry. Journal of Functional Analysis, 220 (2005), 265–303.
J. Brodzki, G.A. Niblo, N.J. Wright, Property A, partial translation structures and uniform embeddings in groups. ArXiv.
J. Brodzki, S. Campbell, E. Guentner, G.A. Niblo, N.J. Wright, Property A and CAT(0) cube complexes. Preprint