A DS set to Auto mode will boot to the cartridge (and you can reflash the firmware to skip the health and safety screen). From there the OS is replaced with whatever is on the cart. A flashcart with the right shell will boot right into whatever app you want (and you can soft reset the console with a key combination to switch apps).

3DSes require a little more work and have a longer boot chain, but it's been thoroughly broken all the way to the bootstrapping process so you can use whichever firmware version and whatever patches you like with enough effort.

Once a DS has been flashed (skips the health and safety screen) it also disables signature verification for DS download play, so you can beam homebrews directly to your DS' home screen with a wifi card. But this is an awkward process that most people don't actually do with their original DSes, as it requires putting tinfoil over a toothpick and jamming it into a hole next to the battery to close the flash write jumper. I think DS' crypto has also been defeated but I can't find any documentation of arbitrary download play on unflashed DSes. Also seems no .nds signing keys in the leaks from what I can tell.