FADC: what is it excactly, and how does it work?

Many moves in Street Fighter IV can be FADC'ed, and it's a cornerstone of higher-level gameplay. Often, they're used to make otherwise-unsafe moves safe, or to extend combos. For many characters, the only way to incorporate an Ultra into a larger combo is to use an FADC to make it possible.

Cancelling any move (normal or special) into a focus attack costs two bars of super meter. You can identify this visually as the character will flash yellow briefly; note that this is not the same thing as the yellow flash that occurs when an EX special move is performed. While cancelling into a focus attack and then simply unleashing the focus attack is possible, in practice, this is almost never done. Generally speaking, the focus attack is almost always immediately cancelled into a dash (forward or backward), hence the name "Focus Attack Dash Cancel." If your goal is to escape from pressure and retreat, you might FADC and then dash backwards; if you want to apply more pressure and/or possibly combo into something else if the first move actually connects, then you would dash forward.

Some of the simplest moves to FADC are Shoryuken-like moves, including Sagat's Tiger Uppercut, Adon's Rising Jaguar, and even Guile's Flash Kick. With a few exceptions, almost all of these sorts of moves can be FADC'ed, though the timing will sometimes vary based on the strength of the move you're using. Aside from Shoryukens, many other sorts of special moves are commonly FADC'ed, either to allow for combo followups or to make them more safe (such as Zangief's Banishing Flat).

In these cases, all you really need to do is perform the move, then, just as the move comes out, press MP+MK and then immediately execute a dash. With a bit of practice, you can learn to execute it as one single motion. In general, if you're having trouble doing it, it's most likely that you're not executing the FADC fast enough, as it needs to be done pretty quickly. Akuma's HP Shoryuken against a blocking opponent is a decent one to practice the input with, as it hits three times, giving you a large window in which to execute the FADC.