Does Christianity allow for believing on someone else's behalf? In 1 Corinthians 7, verse 14 (ESV) we can read:


  14 For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy.


But in verse 16, Paul goes on to say:


  16 For how do you know, wife, whether you will save your husband? Or how do you know, husband, whether you will save your wife?


So, what exactly does "holy" mean? Is it the same as "saved"? If not, what is the difference?

Does verse 14 refer to the relationship between husband and wife, and the relationship parents have with their children in particular, or is it better understood as "any believer in a househould will make all other members holy"?

Are there other bible verses about this topic that might help clearing this up?

Apparently, Acts 16, verses 31 to 34 are similar:


  31And they said, “Believe in the Lord Jesus, and you will be saved, you and your household.” 32And they spoke the word of the Lord to him and to all who were in his house. 33And he took them the same hour of the night and washed their wounds; and he was baptized at once, he and all his family. 34Then he brought them up into his house and set food before them. And he rejoiced along with his entire household that he had believed in God.


It's not clear to me what exactly the jailer did and what his family members did. Sometimes it sounds like the jailer's action were sufficient to save his family. On the other hand, I don't understand if, for example, the jailer would have been saved if his daughter (or maybe even his sister or other relative) had started believing in Jesus.

If there are different views depending on denomination, I would be interested in hearing them.
No, the Bible doesn't teach that one person's faith automatically saves others in your family/household. Rather, it teaches that it will be more likely for someone to be saved if they have a family member who can assist them in acquiring and maintaining strong faith.

Parents can show their children the way to life by instructing their children in God's wise commands:


  Proverbs 6:20-23 (ESV) - My son, keep your father's commandment, and forsake not your mother's teaching. 21  Bind them on your heart always; tie them around your neck. 22  When you walk, they will lead you; when you lie down, they will watch over you; and when you awake, they will talk with you. 23 For the commandment is a lamp and the teaching a light, and the reproofs of discipline are the way of life.


A person is not accountable for the decisions of his father or son:


  Ezekiel 18:20 (ESV) - The soul who sins shall die. The son shall not suffer for the iniquity of the father, nor the father suffer for the iniquity of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself.


In the experience at Acts 16:31-34, the jailer wasn't the only one who put faith in Jesus. His whole family did. However, the jailer was the first one of his family to put faith in Jesus, and he brought Paul and Silas to the rest of his household so that they would also learn and acquire faith. If the jailer had not believed in God, the rest of his household would not have heard the life-saving message at that time (or maybe ever), so they rejoiced that the jailer had believed.

Being holy is also different from being saved. The Israelites were holy, because God had chosen them to be his people and to give his commandments (Deut. 7:6), but they were not saved from their enemies unless they were trusting in God and being obedient to his commandments.