An Apostolic House is firstly "Apostolic" in that it is governed by or submits (through relationship) to the apostolic grace gifting, allowing this fundamental gift to establish God’s order in the assembly and impart this dimension of grace to their lives.
It is furthermore a "House" implying that it is a dwelling place, a household, wherein members live; a family in which there is a spiritual father, mother and sons.
As an apostolic house we therefore simply acknowledge these two things as being foundational for our understanding of what it means to be the Household of God, namely that:
God has placed a set man (leader) over us who governs with a certain dimension of the apostolic grace (which is part of the five fold ministry grace giftings given by Jesus at His ascension - Ephesians 4:11),
We are part of God’s household as sons and that the set man is the spiritual father of the house, so chosen and ordained by God to be His representative in governing the local assembly. As in the natural we are part of families with fathers and mothers, so also in the spiritual; this is the order God has chosen for His church.
Very importantly also, it is the place where sons of God dwell together; where they are immersed and baptised into the name and character of God, and where Christ comes forth through them by the washing of this Word and the feeding from the very words that come from the Father's mouth, as Christ is being revealed through the flesh.