The most common meaning of "house" is a small building where people live. Use this word when you want to talk about the physical building. The plural is houses.
A home is a place where people live. Use this word when when you want to talk about living there and feeling comfortable there. The plural is homes.
The expression "on the house" refers to something different: a bar or restaurant. That business pays for the drinks. The drinks are given free to the customers, probably because they regularly eat there and pay for their meals.