The specific details vary on the implementation, but the analogue to mining in a Proof-of-Stake system is this: newly minted coin is distributed as a reward to

The specific details vary on the implementation, but the analogue to mining in a Proof-of-Stake system is this: newly minted coin is distributed as a reward to whomever wins the block validation "lottery" for a given block. A validator is required to lock up a certain amount of coin before they can start validating (their stake in the system) and based on the more coin you lock up/the longer you've been locking it up/any other variable depending on the implementation, the greater the chances you'll be selected as a given block's validator, and if consensus is achieved, you get your reward. If your node goes offline or collusion is detected, you get penalized by losing a part/all of your stake, again, depending on the exact implementation.