Simply put, the R4 3DS is an adapter that allows you to increase functionality on your console by allowing you to add a micro sd or sdhc memory card, to store and playback media files.
The R4 3DS is a really the third generation of the R4, replacing the earlier R4 and R4i cards, the original R4 was designed for Nintendo Ds and DS lite consoles, later replaced by the R4 SDHC which added SDHC memory card functionality. Then came the R4i SDHC which was created for the Nintendo DSi and DSi XL consoles, this card, also from the R4i-SDHC.com team, offers support for the latest v1.4.5 firmware. Finally the R4 3DS was released, which offers support for all previous consoles, aswell as support for the latest firmware on the newest 3DS, 3DS Xl and 2DS consoles (currently v9.8.0-25).
The R4 3DS was updated to RTS functionality (real time save), which allows you to save in game progress, and continue from where you left off last. This is a huge improvement as it removes the need to start a new game each time to boot up your console.