Local play is a form of multiplayer which doesn't require any connections to a network or the internet. It is hosted on one machine and either all players are visible in one scene (usually 2D games) or the monitor is split into multiple screens (3D games). Players will require additional input devices to join the game (e.g. Controller).
Hot-seat is a form of local multiplayer where the players alternate between each other. Everybody uses the same keyboard or controller at a time. It is mostly used in turn-based games.