Game Modes control exactly how a round will play, and decide whether it'll be a relatively peaceful shift or an instant disaster with no survivors.
Traitor is the default game mode, which selects anywhere from one to seven of your fellow crewmates to be syndicate agents, with the goal of completing three or four objectives.
Extended is the default fallback game mode, chosen when either voted for or if Traitor could not be started. In extended, you're free to do as you wish with no outside antagonists beyond the usual random events.
Removed as of 1/28/2022
Suspicion is a test game mode from earlier in the game's development, designed for more than five players and with a short time limit. If you've played TTT, it's similar to it, but with more guns and less fun.
Sandbox gives players the ability to use some of the game's debug tools, namely the spawn entities and spawn tiles panels, allowing you to build whatever you want using the large number of objects the game contains.
Secret is the game mode that randomizes the current mode and conceals it from crewmates until discovered. The modes in it's selection pool are Nuke ops, Traitor, and Extended.