I'm not aware of any other consensus algorithm that is proven to be as secure as PoW.
The way it works is that "validator" nodes confirm the blocks. The validators are randomly chosen based on the percentage they have staked.
You stake by owning the coin and locking it up using a staking mechanism tied into the protocol. This locks up your coins for a set period of time.
If a validator does not validate correctly or fakes the transactions then they run the risk of being "slashed". Slashing means they forfeit what they have invested.
Usually you need to stake $10K+ USD but there are also pools where people can combine their money.
Both pools and proof of work mining farms suffer from large groups consolidating the validation power.
One of the newer blockchains (Caradano) introduces the concept of a penalty for large pools in favor of having lots of smaller validators to make the network more resilient to a single group having too much validation power. This encourages decentralization.
The fact is, I see all of those massive distributed consensus algorithms as much more robust than they appeared initially. 51%/double spend attacks in practice are extremely hard to occur because of the exposure/publicity of the ledger and how hard it would be to capitalize on crashing the currency (PoS/PoW does not change it).
They're not mathematically provable and the jury is still out. Attackers who know some exploit might be waiting for the much larger Ethereum to switch to PoS so they can reap maximum profit.