Slashing is an automatic penalty in Proof of Stake. If a validator breaks the protocol, the chain takes part of its stake. The threat of that loss is the security. You get paid to validate. You lose money if you cheat.
Slashable acts are the ones that can split or rewrite history. Double-signing means proposing or attesting to two conflicting blocks at the same height. Surround voting means new attestations that contradict older ones in a way that could enable a long-range attack. Some chains also slash long downtime.
When a validator is slashed, some or all of the stake is destroyed, and the validator can be kicked out of the set.
Ethereum's penalties are large. The first hit is at least 1 ETH. Then a correlation penalty grows if many validators are slashed at once, which is the signature of a coordinated attack. If enough validators misbehave together, the penalty can take the entire stake. That is the point: a large attack should be ruinous.
You can only be slashed for messages signed by your validator key. False accusations do not work without that signature. Accidental slashing still happens. The usual mistake is running two validator instances that double-sign. Setup matters. Slashing is the stick next to staking rewards. It makes attacking expensive even for a well-funded adversary.
Ethereum slashes validators who double-sign or surround-vote. A portion of stake is destroyed. That is the penalty that makes proof of stake costly to cheat.
Slashing Mechanism
Interactive simulation of how validators are penalized for malicious behavior in Proof of Stake networks