Free standing stoves are units which sit out away from a wall, not inside it. Usually cast iron, free standing stoves have a door and damper allowing for some control over heat and airflow. Smoke is vented through a pipe going through the ceiling and out of the roof, without a typical chimney.
Open fireplaces are the most traditional way of heating your home with a controlled fire indoors. Open fireplaces are usually brick or rock with a damper in the top leading up the flu and out the top, preferably through a cap.
Inserts slide inside of a traditional masonry fireplace. Inserts are usually installed to do two things: to control the flow of oxygen to the fire and to help more effectively disperse the heat from the fire. Inserts completely enclose the fire with doors and a damper, similar to a stove. Inserts should have a metal liner attached to the top that terminates at the top of the fireplace flue. If your insert does not have this we recommend not using it for safety reasons.