3DS:Flashcarts

There are two kinds of flashcarts that the 3DS can use: DS-mode (such as the R4 family) and 3DS-mode (such as Gateway-3DS). DS flashcarts are still sometimes used in custom firmware setups (though nds-bootstrap is usually a better option, so ntrboot is a more practical use of the cart in those cases), while 3DS flashcarts have fallen out of favor due to CFW entirely replacing their functionality.

DS flashcarts

DS flashcarts refer to any flashcart that runs in DS mode. While such flashcarts may have 3DS in the name (e.g. "R4i SDHC 3DS RTS"), they do not support 3DS games. You can verify this by seeing if the cartridge supports the DS family of consoles (DS / DS lite / DSi). If it does, then it is a DS-mode flashcart.
Most modern DS flashcarts work on the 3DS just as they did on the DS and DSi. Luma3DS re-enables flashcarts that were blocked through 3DS updates, such as the Acekard 2i. The flashcart blacklist has not been updated since system version 7.x (released in 2013), so any flashcart made after that year should work on unmodified consoles, even on the latest firmware.
Old flashcarts that never worked on 3DS or DSi firmware (such as the original R4) can still be used on the 3DS with some workarounds.

3DS flashcarts

3DS flashcarts have been entirely superceded by custom firmware setups. Even if you have an old 3DS-mode flashcart lying around, you should not use it; they are at best cumbersome and at worst a potential brick risk if you're not very careful. No support on flashcart-based setups will be provided on this wiki or by affiliated support channels except for the purposes of migrating people off of flashcart-based setups.

There were three major 3DS-mode flashcarts released over the lifespan of the 3DS: Gateway-3DS, Sky3DS, and Stargate. Any other flashcarts are clones of one of these cartridges (most likely Gateway-3DS).

Gateway-3DS

Gateway-3DS was the first flashcart for the 3DS, and one of the earliest ways to run 3DS games from a flashcart. It required its own custom firmware to function, generally relying on an out-of-date SysNAND (often on 4.x or 9.x) and an up-to-date EmuNAND. Most 3DS flashcart clones are based on this cartridge.
Gateway-3DS came with two flashcarts, one blue and one red. The blue one was just a traditional DS flashcart that was occasionally required to install an exploit to run custom firmware, while the red one was the 3DS-mode flashcart.
The cartridge/custom firmware is notorious for its implementation of a clone flashcart detector which could trigger at random for genuine flashcarts, and when triggered, would cause a software brick in the 3DS. This was notable for being the only software brick that requires a hardmod to fix, but recent developments have created a way to fix gateway bricks via ntrboot. For assistance with this fix, join the Nintendo Homebrew Discord and ask, in English, for help.
Gateway's custom firmware no longer works properly on the latest firmware and is generally not compatible with boot9strap-based setups. If you have such a setup, you can flash NTRBoot to the blue cartridge and use it to install boot9strap.

Sky3DS

Sky3DS is the other most common 3DS flashcart. Unlike Gateway, it doesn't require modifications to the console to use. Sky3DS cartridges work by pretending to be a retail cartridge and cycling through games through a physical button on the cartridge, so they cannot be used to directly run homebrew applications (they can, however, be used to run *hax through cartridge-based exploits).
Because Sky3DS is not a legitimate cartridge, it does not have the private headers that are available on a genuine cartridge. Users of Sky3DS cartridges may therefore get banned from online play unless they get a donor header from a genuine cartridge, but due to the nature of private headers the same header can be used for all games ran through this method.
Sky3DS cartridges still work on the latest firmware, but because of how cumbersome they are, custom firmware is considered to be vastly preferable.

Stargate

Stargate was a short-lived 3DS flashcart released in around 2018. It was most likely developed by the same team between Gateway-3DS. The main feature of this flashcart is that it offers three functions in one: it is an NDS flashcart that is capable of ntrboot with Sky3DS-style 3DS cartridge emulation for playing 3DS backups.
Using its ntrboot feature, Stargate offered its own boot9strap-based custom firmware installation pack with generally weird payloads. If you have such a setup, you can probably migrate away from it by wiping your SD card and following Updating B9S. Otherwise, the cartridge can still serve as a DS flashcart / ntrboot cartridge. The 3DS cartridge emulation should be avoided for the same reasons as Sky3DS cartridges.