Irregular shape. Astigmatism refers to a lens that is not perfectly round. Our corneas and natural lenses appear round to the naked eye, but are in fact not perfectly circular (soccer ball shaped). The more football-shaped they are, the more astigmatism they have, and the more distorted things appear through them. Glasses and contact lenses correct for the astigmatism, removing the blur and distortion it causes.
Corneal shape. Astigmatism is a difference in the refractive power of the eye surface. Usually the corneal surface is like a sphere, think beachball or basketball. With astigmatism the corneal surface is more like a football with different curves in each direction. You can get astigmatism from other structures, such as the lens, but this is much more rare.