Albert Barnes Commentary Acts 16:31

"And they said, Believe on the Lord Jesus, and thou shalt be saved, thou and thy house." — Acts 16:31 (ASV)

Believe on the Lord Jesus Christ. This was a simple, a plain, and an effective direction. They did not direct him to use the means of grace, to pray, or to continue to seek for salvation. They did not advise him to delay or to wait for the mercy of God. They told him to believe at once; to commit his agitated, guilty, and troubled spirit to the Savior, with the assurance that he would find peace. They presumed that he would understand what it was to believe; and they commanded him to do the thing. And this was the uniform direction which the early preachers gave to those inquiring the way to life (see Barnes on Matthew 16:16).

And thy house. And your family. That is, the same salvation is equally adapted to, and offered to your family. It does not mean that his family would be saved simply by his believing; but that the offers had reference to them as well as to himself; that they might be saved as well as he. His attention was thus called at once, as every man's should be, to his family. He was reminded that they needed salvation; and he was presented with the assurance that they might unite with him in the peace and joy of redeeming mercy . It may be implied here that the faith of a father may be expected to be the means of the salvation of his family. It often is so, in fact. But the direct meaning of this is, that salvation was offered to his family as well as himself; implying that if they believed, they should also be saved.

The word "house" can also mean "household."