Payment Requests are a proposed new way to make Bitcoin payments.
Instead of sending somebody a bitcoin address and asking them to pay to that address, you give them a PaymentRequest message which bundles up more information than is contained in just a bitcoin address. A SignedPaymentRequest bundles a PaymentRequest with a digital identity and signature, so you can be sure that you are paying the correct person or merchant.
See https://gist.github.com/4120476 for the current specification. After some prototyping it is expected the specification will become one or more BIPs.
To avoid conflicts, implementations should use this page to register extensions to the protocol.