An economical option for outdoor or feral cats involves a plastic storage bin, a cardboard box to fit inside, and Styrofoam squares for insulation. You can place a vinyl flap over the door to keep out wind and rain, and place towels and blankets inside for comfort.
You can build a fancier cat house using wood. You can place these houses indoors, outdoors or in outdoor cat enclosures. If you plan to place the house outdoors, you should treat the wood to withstand weather and termites. You can elevate the house to provide exercise or cover it with carpet or other materials for scratching, and add decks for the cat to lounge on. Some municipalities require permits for building outdoor cat houses.
It is also possible to make houses for indoor cats with cardboard boxes and t-shirts. You can cut a hole big enough for the cat to use as a door in one side of the box, and turn it upside down. Alternatively, you can use a 15 x 15-inch piece of cardboard, two wire coat hangers and a medium sized t-shirt. Tape the hangers at either end of the cardboard, and stretch the shirt over them. The cat can use the shirt head opening as a door.