Overgrazing is the act of allowing animals to repeatedly go back
to the same plants over a short period of time--generally without
any significant amount of rest or recovery--forcing those plants to
reduce root or below-ground biomass and above-ground biomass.
Overgrazing is a factor of time and poor management, so while
overgrazing is bad, severe overgrazing is what will guarantee
significant soil erosion and loss in plant and soil health.
Severe overgrazing is when plants are grazed so close to the ground that soil is exposed. When soil is exposed, it has greater potential to be blown or washed away by winds and rain, respectively. Severe overgrazing can be caused by any grazing domestic animal species, however it should be noted species like horses and sheep are the worst contenders because of their dextrous lips enabling their teeth to get closer to the soil surface.
Note as well that overgrazing, severe or not is actually not the worst cause of soil erosion. With overgrazing plants are still rooted to the ground and still anchoring most of the soil to the earth. However with crop production, particularly conventional tillage where soil is exposed prior to seeding and tilling after harvest, no soil is being anchored down by any plants so it is much easier for the winds and rain to take it and carry it away to areas where it will never return.
Just like with better managed grazing practices where animal movements are controlled and pastures are divided into paddocks to allow for more rest, crop fields can be better managed with no-till cropping or reducing the amount of tillage done. For instance, fields don't need to be tilled up almost immediately after harvest, they can be left fallow over winter and tilled somewhat lightly before seeding. It should be noted that we cannot prevent all forms of soil erosion, but we can create better management practices to reduce it.