What does Acts 18:8 mean?

The synagogue in Corinth has heard enough from Paul. He's likely had several weeks–or months—to explain to them how Jesus is the Messiah they've been waiting for—how Jesus' life is spelled out in the prophecies of the Hebrew Scriptures. His message is too much for them. They oppose and revile him, and he responds by relieving himself of responsibility for them (Acts 18:6).

Fortunately, some do believe him. First, Stephanas and his family (1 Corinthians 16:15). The Gentile God-fearer Titius Justus not only believes, he offers his home, next door to the synagogue, as an alternate meeting place. Crispus, the ruler of the synagogue, follows.

"Believed in the Lord" means that Crispus believes that Jesus is the Son of God who came to earth as a human and died to cover his sin. Crispus believes that Jesus' resurrection proves that God accepts Jesus' sacrifice, and that it is Jesus' work, not his own, that will reconcile him to God.

Crispus' story provides a testimony against household salvation: the belief that if the head of a household is saved, the rest of the family automatically are, or, that if parents are saved, their children are automatically saved, as well. When Paul and Silas explain the gospel to the jailor in Philippi, casual reading of the passage suggests if he believes his household will be saved (Acts 16:31). There are other passages where the head of the house believes in Christ and the whole family is baptized (Acts 16:15). Here shows that Crispus and his family members believe. Baptism isn't even mentioned, although, later, Paul will remind the Corinthian church that Crispus is one of the few people Paul baptized (1 Corinthians 1:14).