PoW = Proof of Work

Miners compute hashes to solve blocks and get coins.

PoS = Proof of Stake

Your wallet balance determines your chances of solving a block and getting coins.

I'm not sure about how other PoW + PoS hybrids work, but Peercoin is an example of a coin that is initially PoW but gradually transitions to a PoS stage over time. In the beginning, coins are produced via traditional PoW mining just like the way Bitcoin is mined. Over time, PoS begins to overtake PoW until the majority of new coins are generated via PoS minting. Peercoin uses coin age which means that coins must be kept in a wallet for a certain length of time before proof of stake starts to take effect. Furthermore, PoS coins usually require you to keep your wallet running and unlocked in order to stake coins.