Wooden doors can easily have pet doors installed in them. Read the directions that come with your pet door to determine the size hole that needs to be cut. For hollow core doors with flat faces, you can simply cut the hole in the right place. If you have a door with raised panels in it or other uneven surfaces, you may need to infill with bits of wood or with caulking to create a flat surface around the edges of your pet door.
Metal doors are usually made with thin metal facings on the interior and exterior and either wood or foam insulation inside of them. Pet doors can be installed in metal doors in the same way as wood doors except that you'll need a different saw blade or a pair of tin snips to cut through the metal facing. Be certain that the frame of the pet door covers any cut metal edges completely before letting your pet use the door to avoid the risk of injury.
Glass doors that are made with one large sheet of glass in a frame can't have a pet door installed in them. Glass doors that have decorative mullions on them (the thin strips of wood in between panes of glass) also can't have a pet door installed because the glass inside of the mullions is one large sheet. A door that is built with real mullions and numerous small panes of glass can have a pet door installed by removing one of the small panes of glass and replacing it with a pet door.
Some pet door companies make pet doors that are designed to fit into screen doors. The pet door is usually a plastic frame with a hinged screen door in it. Normal pet doors can't be used in screen doors because their frames are designed for solid doors and are too wide to clip to a screen. Also, the weight of a normal pet door would probably be too heavy to set into a screen. Screened pet doors allow your animals to come and go in the summer without bringing mosquitoes and other insects in with them.