Proof-of-replication (PoRep) is an interactive proof system in which provers defend a publicly verifiable that they are dedicating unique resources to storing retrievable replicas of a data file. It functions as a proof-of-retrievability mechanism embedded within a proof-of-space mechanism (a type of proof-of-capacity).
PoRep enables a prover to demonstrate that they're using some minimum amount of space to store information (as in proof-of-space) as well as to actually use that space to store useful information (unlike most proofs-of-space). At the same time, PoRep enables any data that is being stored to be extracted efficiently (as in proof-of-retrievability). Peers in the network are incentivized to participate as provers and to store data files because doing so enables them to earn network rewards.
One important property of PoRep is that there is no economic incentive for a prover not to store replicated copies of data files as they are claiming to do. This is achieved by designing the system such that only the size of the data inputs matters to a prover's success in the protocol, meaning that it costs the same to store any number of duplicated data files as it does to run the protocol on the same number of distinct data files.
PoRep schemes were designed for use in cloud storage and decentralized storage networks such as Filecoin. In such networks, it's important to ensure that replications of data files are indeed being stored independently.