Recoupment is the practice, common in the music industry, of claiming an advance provided to an artist back from that artist rather than (or, as well as) from related sources. Many record labels practice recoupment, and most new artists have little choice but to accept it since they lack the negotiating power to obtain a better contract.
Recoupment is also a common practice within the insurance industry. It means virtually the same thing, but generally occurs when an insurance company has paid the benefits to a hospital and then takes back that money when they have determined that those benefits should not have been given.
Suppose that a label gives a band $250,000 to record an album. They agree to do so in return for 90% of the sales. The album is recorded, and sells 200,000 copies at $10 each, yielding $2m. The record company takes 90% of this as agreed, leaving the band with $200,000 of their own. This is the situation without recoupment.
With recoupment, the label gives the band $250,000 as before. The album again sells 200,000 copies at $10 each, yielding $2m. The record company takes 90% of this as before, leaving the band with $200,000 - but the record company no longer considers the 90% of sales as repayment of the advance. Instead, the band is required to pay the record company back the advance from their own cut, leaving them in $50,000 debt to the record company - even though the record company has gained $1.8m from the sales percentage.
Suppose a patient has visited the hospital and has given the hospital their insurance information. Once the visit is complete the hospital will bill the patient's insurance company. The insurance company will then pay based upon a determined amount of benefits due the patient. Many times insurance companies, especially in the case of Medicare and Medicaid, will later determine that the patient was not eligible for those benefits and will recoup (take back) that money from the hospital.