"Proof of Capacity" is also abbreviated with "PoC" and is a still little-tested consensus algorithm (and yes, it would be a competitor to Proof of Stake and Proof of Work). To "mine", you must allocate space on your hard drive for the process, and the more space you allocate the more chances you have to find blocks and get rewards. It is also called "Proof of space".
The only cryptocurrency I know that is using a public blockchain and this algorithm is Burst, but there are other, even more advanced proposals (see this Wikipedia article and this proposal from Bittorrent creator Bram Cohen).
Good call -- It is much more likely to see a Proof of Capacity project than Proof of Concept. I probably should have considered explaining that one as well in case he was confused with the original question.
I personally believe PoC (capacity) will be much more regularly utilized in the future. There are a lot of up and coming projects that revolve around decentralized storage.