
I think most people here know what the Famicom Disk System is, but rarely is it actually discussed in depth. So I'm making this thread to drop what little knowledge I have and to hopefully generate discussion. A couple of images here were yanked right from the Racketboy NES/Famicom article. I await my ban.

The Famicom Disk System is a peripheral that attaches to the Famicom
Released in 1986, the FDS is attached to the Famicom by way of a RAM adapter that plugs directly into the cartridge slot. The FDS can be run on batteries as well as a traditional AC adapter; the battery option was included as Nintendo figured that a standard outlet would already be consumed by the Famicom and television set.
The games themselves are housed on proprietary 2.8" floppies. Most games utilize both sides of a single disk (sides A and B) and must be flipped occasionally during gameplay, while some other titles are only one-sided or, alternatively, are large enough to require two disks.
Unlike, say, the Sega and NEC CD-ROM games, FDS games superficially don't appear to be that much different (or "better") than traditional Famicom games. There is a more advanced sound chip in the FDS hardware that some develops took advantage of, but graphically most FDS games resemble their cartridge-based peers.

There's also an all-in-one Famicom/FDS console
The Sharp Twin Famicom is essentially comprised of Famicom and Disk System hardware placed side-by-side in the same casing. A switch on the system swaps back and forth between Famicom and FDS, and a locking mechanism prevents Famicom cartridge games from being inserted while the FDS hardware is in use. If you're interested in FDS software and don't already own a standard Famicom console, buying a "Twin Fami" may be the ideal solution.

Famicom Disk System games come equipped with a save feature
In fact, this is one major reason why the Disk System exists in the first place. By 1986 it was apparent that certain games were too long to be played in one sitting, and password systems were becoming cumbersome and annoying. The technology that allowed saving directly to carts was still in the developmental stage. Taking a cue from computer gaming, Nintendo chose to use magnetic media for the purposes of recording game progress. Every FDS game I've come across allows for a save of some kind, whether it actually saves the player's location in the game or just a high score. Data is saved to the disk itself, rather than a spare blank disk or to the system.

Disks were (and are) cheap, which isn't always a good thing
Disks were inexpensive to mass-produce, and the savings were passed on to the consumer. Disk kiosks found in hobby shops around Japan also allowed players to purchase blank disks and write the games of their choice to them. Note that regular retail disks can also be overwritten.
This thriftiness by Nintendo helped bring about the downfall of the FDS. Piracy was an enormous problem. Disks themselves are prone to failure after repeated use, and the dive belts on the FDS system itself corrode over time.
When shopping for FDS hardware/software today, look for systems that have had belts replaced and assure that the games themselves are in working order (many eBay sellers post screenshots). Also, if this is a concern to you, pay attention the label on the floppy to see if it matches what is actually on the disk.

Many games we know as "NES Classics" were originally released on the Famicom Disk System
Zelda I & II, Castlevania I & II, Metroid, Kid Icarus, and others all made their initial appearance on the FDS. In their transition to the North American NES some arrived with their save feature intact by way of a battery (Zelda), some had the save replaced with a password (Metroid), while others had the save feature scrapped entirely (Castlevania).
Super Mario Bros. 2 also originated on the Disk System, in more ways than one. The original SMB 2 was a diabolically hard nightmare that Nintendo dared not unleash upon the West. Instead a completely different disk game, Yume Kōjō: Doki Doki Panic, was retooled to become to the North American Super Mario Bros. 2. Eventually the first SMB 2 did make its way over here by way of the Super Mario All-Stars compilation (SNES), where it was renamed The Lost Levels.
Certain disk games in Japan also had cartridge re-releases (Zelda Famicom carts finally appeared in 1994!).

The Famicom Disk System library is very diverse and of exemplary quality
There's a surprising amount of quality stuff to be found here. In addition the first-party Nintendo classics, Konami was a heavy supporter of the FDS with many of their best 8-bit titles (such as Ai Senshi Nicol, Almana no Kiseki, and Falsion) appearing as disk exclusives.
The same glut of genres represented on the Famicom can be found here as well - sports, shmup, platformer, puzzle, strategy... The RPG library is incredible, with a plethora of games that are either straight-up FDS exclusives or ports of earlier computer versions. And yes, since this is Japan the requisite amount of (unlicensed) "eroge" material is also available.

The emulation and fan translation scene is thriving
Most NES emulators will run FDS games, though you'll need to find the FDS BIOS first. And yes, the disk images do need to be "flipped" even when playing via emulation.
Numerous games have received fan translations, including most of the notable RPGs.
For those looking for official emulation, Nintendo's virtual console currently has Super Mario Bros. 2 (The Lost Levels) available, as well as the Zelda-on-steroids adventure title Nazo no Murasame Jō.
Widespread piracy, as well as the advent of lithium save batteries, eventually brought the Famicom Disk System down. But, like so many other retro consoles, it still remains exceptionally enjoyable to this day. Time to get flippin'.