To sweat is to cook over low heat in a small amount of fat, usually in a covered pan or pot. Often used to describe the way aromatic vegetables such as onions, carrots and celery are cooked prior to adding other ingredients.
The objective in sweating vegetables is to soften them and release the moisture in them, not to brown them. This release of moisture is how the term "sweat" gets its name. It is also known as butter-steam.
Frequently the fat is provided by rendering fat from some fatty meat, such as bacon, which also adds additional flavor.