What is house visitation?
This is when the elders of the church visit the members of the church to assist them in their life of faith and walk with God.
How often is this done?
Under normal circumstances, the elders aim to visit each family once per year.
Why once per year?
This is a long standing tradition that appears to have begun in Geneva under Calvin's leadership. See Ray Van Neste, “Pastoral Pensées: The Care of Souls: The Heart of the Reformation,” Themelios 39, no. 1 (2014): 57; or here.
What was Calvin's practice?
Beza writes the following about Calvin's practice:
The year 1550 was remarkable for its tranquility with respect to the Church. The Consistory resolved that the ministers should not confine their instructions to public preaching—which was neglected by some, and heard with very little advantage by others—but at stated seasons should visit every family from house to house, attended by an elder and a decurion [a person in charge of ten(?); probably something like a district manager] of each ward, to explain the Christian doctrines to the common people, and require from every one a brief account of their faith. These private visits were of great use to the Church, and it is scarcely credible how much fruit was produced by this plan of instruction. source
Does the Bible ever speak of this practice?
The first instance is in Acts 5. Some of the apostles were arrested and jailed. After their trial, they were flogged and released. Then we read:
The apostles left the Sanhedrin, rejoicing that they had been counted worthy of suffering disgrace for the Name. Every day, in the temple courts and from house to house, they did not stop teaching and proclaiming the good news that Jesus is the Christ. (Acts 5:41-42)
How are we to understand the words "and from house to house"?
Clearly, these were meetings which were taking place in private houses, not in the temple. Whether these were gatherings of families or of a local church (or both) is not clear. From the text, it is plain that the apostles never missed on opportunity to declare the person and work of Jesus.
What other Scriptures mention this practice?
Paul tells the Ephesian elders in Acts 20:
When they came to him, he said, “You know how I lived the whole time I was with you, from the first day I arrived in the province of Asia. I served the Lord with great humility and with tears, especially in the trials that came upon me through the plots of the Jews. I did not shrink back from declaring anything that was helpful to you as I taught you publicly and from house to house, testifying to Jews and Greeks alike about repentance to God and faith in our Lord Jesus Christ. (Acts 20:18-21)
How does this passage differ from the one in Acts 5?
In Acts 5, the purpose of going house to house was more evangelistic. Here the motive seems to be more teaching those who are already Christians as we would expect in Ephesus where the church was more established.
What is the basic reason for this practice?
In keeping with the apostles' practice, the leadership of a church should miss no opportunity to help the members forward in their life of faith and walk with God. Whatever issues they may be facing, whatever questions may be troubling their mind, whatever sins they may struggle against, whatever trials they may be suffering under, with all these things, the elders can come alongside them, extend a supporting hand, and especially bring the light of God's word to bear on whatever they are dealing with. This certainly happens in church but also outside the church. It happens in a corporate setting and in a private and one-to-one basis.