All game data is saved on the game cartridge itself. The DS does not have internal memory for saving game data - the DSi has a memory card, but also does not save game data to it.
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cartridge
Well, a video game works very complexley. The data is all on the disk unless saved to the game console(this can only be done on a playstation 3). When you insert the video game disk into the game console, there is a laser light that reads the disk. The data on the disk is then ejected onto the screen and connected to the game controller ( so you can control your video game person). 3-D games are not really 3 demensional, they just look 3 demensional. they are data specs projected onto the screen and change color when activated by the game controller. And that is how a video game works.
The game ID shold be on the back of the game cartrige.
First you start a new game and then just save over your old saved game.
If you mean a game save [under Game > Saved Data Uility on your XMB] then the games that you play will automatically load those for you. If you mean a saved homebrew game, read the readme.txt they usually come with, it will tell you to move all the data to "psp/game" or maybe "psp/game150" ect..