(Heb. 3:3-4) What is “the house” mentioned here?

The house here is the group of believers throughout human history. The author says that Moses served in this house in verse 5, and he says that all believers are in this house of God in verse 6 (“whose house we are”). Since Moses was a member of this group, he is a part of the house, so to speak.

The author has just told us that Jesus was fully human. Does this mean that he is just another prophet, another servant in the house? No! While “Moses was faithful in all His house as a servant” (v.5), the author tells us that “Christ was faithful as a Son over His house” (v.6). Christ wasn’t a servant in the house (v.5); he was a son (v.6).