There are only two things that a villager can interact with: you, and the house.
Most of the conversational interaction is just fluff and commenting about your home. The villager will mostly wander around a little until the visit is over.
During the short time frame of the visit, one of these furniture-related events may happen:
When a villager visits your home, speaking to the villager may eventually result in him/her making a remark about your home "missing something". This is usually followed by the villager giving you a present that he/she thinks would go well with your home.
When you visit a villager's home, speaking to the villager may eventually result in him/her offering furniture in his/her home for sale. However, certain items won't be sold to you, such as tables with objects on them, or furniture characteristic to the villager's personality/preferences.
During the end of a villager's visit, you also have to answer a question about how appealing the home is. Despite the fact that the villager can be offended by praise, or even be delighted by criticism, this part doesn't actually do anything meaningful.