Action Replay DS:
These devices were usually black with a red and black label on the front. Two branches of models existed at one point, the second models being the latest of the ARDS devices before they went out. Model one devices came with older firmware versions preinstalled, some of which wasn't able or allowed to be updated to any version above v1.55, back in the day two types of model one revisions existed, one with a cartridge slot for game cards and one without a cartridge slot (requiring you to do cartridge swapping). Model two ARDS devices had more support and stability added to them, replacing the predecessor models and support for them, the latest most of second models can run is v1.71. Both the model one and model two variations only worked on the Nintendo DS and Nintendo DS Lite.
Action Replay DSi:
These devices replaced the original Action Replay DS models and came in many different forms due to the numerous amount of updates made to the devices. Most of the versions were pretty much the same, the only difference being the firmware and or sticker they had on the front (black, yellow, and blue color variations existed, they all function the same however). There was one model that stuck out the most, and that was the ARDSi Media Edition, which had an SD card slot for extra functions (fun fact, you can actually put a microSD slot on ARDSi devices that don't have one, you can do this because they use the same PCB as the Media Edition model, they just lack a microSD slot obviously). These devices work on the Nintendo DS, DS Lite, DSi, and 3DS, however these no longer auto boot when turning the system on. This is due to the way Datel went about their methods with getting unlicensed content to run on the systems, instead of using custom boot images they instead opted to spoofed images to bypass any of the signature checks that the newer systems (the DSi and 3DS) would have in place (Nintendo was on a spree trying to block devices like these).
Miscellaneous ARDS Devices:
You may have noticed every now and again an ARDS device that has a label related to a game, like Pokemon for example. Obviously these devices tend to lack a slot to hold the game cartridge, that's because the intended design is the swap game cartridges out after selecting the cheats you want to use (like the old model one versions of the older ARDS devices back then). These devices aren't that special at all really, all the cheat codes they came with you could get right off the internet for a regular ARDS/ARDSi device easily. These funky devices were intended for a smaller audience who didn't want a device for multiple uses, just something simple.