Both "house" and "kingdom" are interesting words because the don't quite mean the same thing in Jesus's Greek as the do in English. Their meaning is important not only to the terms themselves, but to the many different terms related to them.
"House" is not one term in Greek, but two oikia and oikos. One is the female form of the word and the other the masculine. They have very similar meanings though oikos is broader than oikia. Jesus seems to use them interchangeably, but I haven't made a close enough analysis of his use to say for sure. It seems like the different evangelists use these two different terms in similar verses, further clouding their differences.
"House" as oikia, means "house", "building," and "household." It refers to the building itself, all the people that dwell in it, including slaves and servants, all property owned by that family, and all the descendants of the continued line. We might say "estate" in English to capture this idea moe clearly.
"House" as oikos, which means "house", "dwelling place", "room", "home", "meeting hall", "household goods", "substance," and "ruling family." It is any dwelling place but not exclusively a separate house. It is any dwelling place but not exclusively a separate house. It means the household or clan that lives in the building as well. which means "house", "building," and "household."
Several other Greek terms, translated in various ways, are based on this concept of a "house." "Goodman of the house" is from a compound Greek, οἰκοδεσπότης, word that is literally the "master of the house." It was translated as "householder." The verb often translated generically as "build" is a word that specifically means "build a house," oikodomeo, though it generally, "build", "fashion," and so on.
Lest we forget, but Christ and his earthly father Joseph were house-builders so the idea of a "house" had a rich symbolism for him. (The Greek term describing Christ and translated as "carpenter" is actually closer in meaning to "mason" and refers specifically to someone who builds houses as opposed to working with wood, for example, building furniture.)
As a house builder, Jesus uses the terms "house" in both its meanings. He uses it to as a structure whose architecture matters. His first major parable using the idea of a house deals with building it on a solid foundation (Mat 7:24). For him, a house that cannot stand is in a very physical sense, no house at all.
Second, he sees a "house" as a family or a clan, encompassing all the people in the house and headed by the master of the house. For Christ, the world of men is divided into kingdoms and kingdoms are divided into houses (Mat 24:43). Belonging to a given "house" is a matter of personal allegiance (Mat 19:29) to the master. The house can be judged as a group (Mat 10:13). The role of the master of the house is to protect the people and property within the house. (Mat 12:29, Mat 24:43).
In Mark 3:25, Jesus says that a house divided against itself cannot stand. On one level, this verse is a simple logical statement about what makes up a house. A divided house where there is a clear split between its members become two houses. Think of a clan where one branch of the clan separates itself from the others.
two houses. A member of a house cannot be against the master of the house because his membership in a house depends on his allegiance to the master. Nor can a master of a house be against his (or her, widow's also have houses Mat 23:14) house's members because his authority comes from the responsibility of protecting those members from outsiders. The violation in either half of this contact, the leader's or a follower's, destroys the house itself as a unit.
"The kingdom" is from the Greek word basileia, which means "kingdom", "dominion", "hereditary monarchy", "kingly office," (passive) "being ruled by a king," and "reign." The word translated as "kingdom" can be the region, the reign, the castle or the authority of a ruler. Christ does not seem to use it to mean a physical region, so its translation as "reign" or "realm" seems more appropriate. This is especially true because the "reign" of a king means the execution of his will.
The Greek "kingdom" is more specifically under the authority on one man or one house, where in English, the term can mean a nation and a people. The word, like our English one, comes from the Greek word translated as "king," basileus, which means a "king", "chief", "prince", "lord", "master", "a great man," and "the first and most distinguished of any class." This is the top person in a region of any size.
From Jesus's viewpoint, a kingdom has one leader and must be united behind him. Beneath the kingdom, a nation is divided into "houses" where allegiance goes to the master of the house. However, the king has authority over all the houses below him.
Of course, the most common reference Jesus makes to a "kingdom" is the "kingdom of heaven" or, more accuraterly, "the realm of the skies." See this article about that phrase. However, Jesus also refers to Satan, better translated as "adversary" or "suffering" as having a kingdom. See the end of this article for more on that.