The house appears in the first three games, but can only be visited by the player in the first game.
The house consists of three rooms; a kitchen, bedroom, and Luigi's secret basement in which he would write in his diary.
In Paper Mario, the house can be visited by going down a Warp Pipe that is located by the Toad Town sign in the main square of the town.
Luigi watches over the house while Mario is out on his adventure. He can be seen either sitting or standing out on the porch, singing, or hanging around inside the house.