With a PAYE umbrella company you become an employee of the company. Each week or month you instruct them to invoice the agency/client based on your time sheet. They then pay you a net salary after deducting national insurance contributions and income tax on all your earnings.
Umbrella companies were spawned following the introduction of IR35 back in April 2000 and became hugely popular after the Management Service Company Legislation in April 2007. They are seen as one of the last remaining 'compliant' payroll methods, taxing contractors fully under PAYE, after the deduction of allowable business expenses.