@mightysprite is correct, but to streamline a bit:
Keep Furniture: All the building's furnishings go with your household to the next house, in the household inventory. The house they just left will be completely empty.
Sell Furniture: All the building's furnishings stay exactly where they are and your household will get money for them. The house they just left will look exactly the way it did when they lived there (no change).
Evict: Furnishings stay in the building (no change). Not sure if they get money for anything.
(I also often save before moving sims to a new home, in case there are any problems.)
Hope that helps!
