Nothing-at-Stake
A theoretical weakness of early proof-of-stake designs in which validators could back every competing chain at no cost.
In proof of work, supporting two branches splits a miner's hardware between them, so there is a direct cost to hedging. In a naive proof-of-stake design, signing on both branches costs nothing, so a self-interested validator would sign everything and forks might never resolve. Deployed proof-of-stake systems answer this directly by slashing validators who sign conflicting messages, by adding explicit finality so a finalized branch cannot be abandoned cheaply, and by using weak subjectivity checkpoints so a newly syncing node does not accept an ancient alternative history. The term now appears mainly in debates comparing the two families of consensus rather than as a live defect.
In practice
Ethereum's slashing conditions penalize a validator that signs two different blocks for the same slot, which removes the incentive the problem describes.
The common misunderstanding
Nothing-at-stake describes a flaw in naive designs rather than in current proof-of-stake networks, which penalize signing conflicting blocks.