How do eye doctors test for astigmatism?

Topography: The most accurate way is to do a topography - a machine shines rings of light onto your eye and then analyses the reflection. It then creates a surface MAP - sort of a 3d interpretation of all the lumps and dips on your cornea. In the end it tells you how much and where you have astigmatism.

Astigmatism: Astigmatism can be corneal (football shaped cornea)in origin or lenticular (crystalline lens). The first test to give it away is vision testing. Refraction tests quantify and confirm astigmatism wavefront measurement further accentuates values of primary & secondary astigmatism including associated coma. Corneal topography, opd analysis & ray tracing can even differentiate lens vs cornea astigm.