Houses provide army capacity to your city. Each unit has its own weight, and extra Houses allow you to recruit and maintain more units at a time. Once your city capacity is reached, no additional units can be recruited. Keep in mind that your city capacity takes into account all units present in your city, on your wall, in your recruitment queues in the Barracks, as well as in any camps outside your city. At the same time, units being healed in your Hospital or reinforced through your Embassy do not add up to your city capacity until they become available for use (until healing queues / embassy orders are completed). You can go over your city capacity by using Spheres of Summoning to get extra units or by healing units in the Hospital once your city is at maximum capacity. This will put you over your city capacity, and while no additional units can be recruited, you will suffer no penalties.
Upgrading each House increases the amount of capacity points it provides. It's recommended to build 10 houses first and keep them as upgraded as you can, as your army is the main "resource" in the game. Additional resources can be mined from the Outposts on the global map.
Required for: Barracks (and subsequently Wall, Hospital, Watchtower and Citadel)