And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
Note 3 at Ac 16:31: Paul and Silas' answer to the jailer about what he must do to be saved is exactly the same as Jesus' answer to the Jews in Joh 6:29. Faith in what Jesus did for us is the only requirement and the only thing that we can do to produce salvation (see note 2 at Lu 11:42 and note 3 at Lu 11:44).
Paul and Silas did not say that this man had to confess his sins. That is a common instruction given to people who ask this question today, and it can be appropriate if, by confessing their sins, people are simply admitting they're guilty and need a Savior. But the belief that people have to confess each individual sin in order to be saved is unscriptural. What would happen if one was forgotten? Would it not be forgiven? We don't confess our sins but rather confess our faith in the sacrifice of Jesus as the perfect payment for our sins (see note 7 at Mr 1:5).
Note 4 at Ac 16:31: Many people have mistakenly used this scripture as a verse that promises us that our loved ones will be saved as we stand in faith believing for their salvation. That is not what this verse means. We cannot get anyone born again by our faith, even if that person is near of kin. Paul and Silas were simply stating that this jailer could be born again by believing on the Lord Jesus Christ, and the same thing would work for his family if they would believe.
This is not to say that there are no scriptures to encourage us to believe for the salvation of our family members and friends. Indeed there are (see note 1 at Mt 9:38). This is just not one of them.