An auto-save usually happens once you reach a checkpoint. A notification will appear on-screen telling you that the game is saving. In some games, you have the option to manually save even if the game auto-saves. An auto-save always overwrites the previous save.
Auto-saves were created for multiple reasons: So that the player doesn't forget to save or in case the game suddenly crashes or quits for some reason. But they became widely popular with the explosion of console gaming where there weren't enough buttons for quick saves or loads so they streamlined the experience. This also allows the player to stay immersed in the game and not have to worry about saving their progress.
Auto-saves happen when the player does things like enter a new area, reach a checkpoint or beat a boss or level. Some games give you the option to turn auto-save off as it can sometimes happen when you are really low on health/ammo or you just want to have more control. Some games have yet to implement the auto-save feature, like Pokemon.