It is a little confusing, but basically light isn't refracted (bent) evenly and to a single point. There are different localized areas of steepness that affects how light will be refracted. The result: rather than light coming to a single point, it is spread out over a 3 dimensional space--the image you see is blurry. The total astigmatism of the optical system--your eye--is the sum of the astigmatism at the cornea and at the lens. During cataract surgery, when you remove the lens, you are left with just the corneal astigmatism. This is why your surgeon may recommend inserting an astigmatism correcting intraocular lens--the toric lens or the light adjustable lens. These special lenses will counteract the residual corneal astigmatism so that you will have a sharper uncorrected image post-operatively. Note that astigmatism can also be corrected by glasses or contact lenses, though it is generally recommended to correct as much with an intraocular lens as possible.

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The corneal astigmatism comes in two different varieties: Regular or Irregular. 

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Regular astigmatism just means that the steep and the flax axes of the corneal plane are perpendicular to one another. Currently astigmatism correcting lenses are only able to correct this type of astigmatism. To get even more confusing, there are also two types of regular astigmatism, called "With the Rule" or "Against the Rule" astigmatism, and this has to do with whether the steep axis of the cornea is closer to 90 degrees or 180 degrees. "With the Rule" Astigmatism is usually better tolerated and tends to get less as we age.

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Irregular astigmatism is when there steep and flat axes of the cornea are not perpendicular to one another. This type of astigmatism can not currently be treated with intraocular lenses.