Proof-of-Weight is a broad classification of consensus algorithms based around the Algorand consensus model. The general idea is that where in PoS, your percentage of tokens owned in the network represents your probability of “discovering” the next block, in a PoWeight system, some other relatively weighted value is used. Concrete example: Filecoin’s Proof-of-Spacetime is weighted on how much IPFS data you’re storing. Other systems could include weights for things like Proof-of-Reputation.
Incentivization can be a challenge
ALGORAND: The Efficient and Democratic Ledger. Micali S. '16.