When a patient presents an insurance card, the information is entered into the computer, and a claim is sent to the insurance company for payment or rejection. What is the process of the insurance company accepting or rejecting a claim called?

Adjudication is the term used for the process by which claims are rejected or paid. This is part of the third-party payor process.
Adjudication is the process of reviewing and evaluating an insurance claim to determine whether it is covered under the policy and the amount of benefits to be paid. The insurance company will review all of the information submitted by the patient and provider, including the medical records, to make a decision.