According to Surgical Specialties Corporation, absorbable sutures can take anywhere from 10 to 90 days to be completely broken down and absorbed by the body. They are produced with different materials that have different strengths and absorption rates.
They are made from natural materials that the body can decompose, such as catgut or collagen, and from newer synthetic materials such as polyglycolic acid.
When a physician or dentist uses absorbable stitches, they expect the wound to be almost entirely healed by the time the suture material breaks down. According to Wikipedia, they are generally used so that patients do not need to return to the doctor just to have the sutures removed, which can be inconvenient and painful. They are also used to hold internal wounds closed. Despite being made of materials the body can break down, they are still sometimes rejected by the body.