The availability problem
A strict quorum requires W of the designated home nodes for a key to be reachable. During a network partition, too many home nodes may be unreachable, and writes would have to fail.
The sloppy version
A sloppy quorum keeps writing anyway. If the usual home nodes are unreachable, the write is accepted by other reachable nodes that are not normally responsible for that key. The write count W is satisfied, just not by the right nodes.
Hinted handoff
Each substitute node stores a hint recording which home node the data really belongs to. When the partition heals and the home nodes return, the substitute forwards the data back to them. This process is called hinted handoff.
The cost
A sloppy quorum boosts write availability but weakens the read guarantee. During the partition, a read of the proper R home nodes may miss writes parked on substitutes, so freshness is no longer assured.
Key idea
A sloppy quorum trades the freshness guarantee for higher write availability, using hinted handoff to return data to its home later.