SS1 saves only your progress (the guys you defeat) and does so automatically if you have a card plugged in and you choose to allow card saves. So if you get up to Amakusa, you can choose any character when you restart to start off on his stage.
Other games vary. Burning Fight automatically saves the last stage you completed. Baseball Stars Pro saves the last inning you played, and so on. It's more of like a password feature than the kind of memory saves we're used to with today's consoles.