On a proof-of-stake network, validators are responsible for proposing and verifying new blocks of transactions. To participate, a validator locks up (stakes) a quantity of the network's native asset as a financial commitment to honest behaviour, and earns rewards for doing the job correctly.
Validators who act dishonestly or fail to perform reliably can have part of their staked funds destroyed, a penalty known as slashing — this economic incentive is what's meant to keep the network secure without needing a central authority.