Your progress is saved automatically at key moments in singleplayer mode: when a fight ends, when a new act begins, when a character levels up, when you buy equipment, etc., but no more than four times per minute.
For every Nintendo profile there are 4 slots for save files in STORY mode. You can see them by selecting STORY from the main menu (the first point from the top).
Every slot displays its current act and character level. Every slot can be deleted by following instructions that appear on the screen (when you press +). If the player changes the Nintendo profile (in the system or when starting up the game), the save slots will be replaced by those on the selected profile. The Nintendo profile selection window at startup can be turned off in the system settings.