Geometric structures can be described globally without reference to coordinates. However they can always be expressed locally in a coordinate system. In the 18'th century and maybe earlier I am not sure, geometers described curves and surfaces by parameterizations and then deduced the geometry from the parameters. This is a coordinate approach to geometry. But later is was realized that geometries exist as global structures across an entire surface and that the surface can not necessarily be described with a single coordinate system.
I can describe examples if you like.