A delivery receipt is the status message returned by the carrier for an SMS, reporting that the message was delivered to the handset, failed, or expired, sometimes with an error code.
After a message is accepted, the carrier's SMSC reports what happened to it: delivered when the handset acknowledged, failed with a reason when it could not, expired when the validity period ran out. This report is the delivery receipt, DLR in telecom vocabulary. It is what turns "sent" into "delivered" in a status page.
Receipts are only as honest as the route. Some operators return handset receipts, some only acknowledge acceptance, and grey routes fabricate them. Knowing which applies per country is essential to reading a report.
At SMSMeteor: receipts drive the message status and the message.delivered and message.failed webhooks; each country page states whether handset receipts or carrier acknowledgement is the final state.