In SNES Java Mode and AOX Score Mode you can only save if the original game had battery saves (also called SRAM). You have to first use the original game's save feature and then press escape and select save. To load you just use the original game's built in load/continue function.
Legend Of Zelda: A Link To The Past, The ยป
Super Nintendo is a game that originally had battery saves and so will save in Java Mode. You can try it to make sure you are doing it right.
To save in all SNES games you have to use AOX Full Mode.
NES, SMS, Game Gear, Genesis and TG16 all support regular save states in all games in Java Mode. But Game Boy and SNES do not.