Fa-jin, according to the Wikipedia article, means "to discharge power explosively". If there is only one way to do this, what is it? If there is more than one way, what are the two or three main ways of doing this?
"Jin" means a trained force, somewhat like a force vector. "Fa" is to attack. In a more practical sense, fajin is a pronounced force that involves the whole body shaking and adding to the power. On a more titillating level, fa jin implies that the body relaxes to produce this burst of force. And it's true. So look for a way to store force and then release it by relaxing. It's an interesting way to approach the puzzle and if you understand it, it will answer a lot of other questions about the evolution of the internal arts and the body mechanics developed over centuries.
發 and 勁 Transliterated
Fa-jin is two characters:
In the context of martial arts, a fair translation of fa-jin would be "emitting force" or something similar.
There be no magic here
Fa-jin is just a Chinese term for explosiveness. This is equivalent to how we say a wrestler might have an explosive power double, or a kickboxer has a powerful round kick, or a boxer has heavy hands. There's nothing special, specific, or mystical about fa-jin. It's just hitting or throwing someone hard and fast.
There are as many ways to employ fa-jin as there are techniques to hit hard with, and the details are fractal. One can go infinitely deep with explaining how to execute a particular technique explosively, but when you recognize the term "fa-jin" as simply referring to the concept of hitting powerfully, giving examples specific to fa-jin becomes obviously fruitless. If you are hitting hard and efficiently, that's fa-jin.
For instance, boxers hit hard (employ fa-jin) by engaging the glutes, twisting the hips, and keeping the trunk rigid to transmit force to the arms, which are as loose and relaxed as possible as they extend. This should sound familiar to hard-sparring exponents of karate, muay Thai, taiji, and so on.
@Wudang tried to explain it from the point of view of a taijiquan practitioner, but in my mind, he mostly just glossed over the description of fajing and of how to get it working.
The term fajin is composed of two words fa, which means "to send out, to issue" and jin is a word used for strength (quite interestingly, one of the pinyin translations I found through Google also mentioned that it meant "crystal", which curiously enough, is fitting in the context).
This means that literal meaning of fajin could be translated "to send out strength" or to "issue strength". And that, quite simply, is exactly what fajin is.
There are probably many methods of issuing strength, mostly these are dependent on the system of MA you are learning. Some of the systems (including taijiquan) generate power for the fajin from the feet, other systems use other methods, that I am not qualified to discuss.
In taijiquan, there is a classic text that describes very clearly how to fajin. Loosely translated (and from my memory rather than a strict citation), it says something like this:
This means essentially, that the act of "releasing the strength" involves your whole body, which must move as one unit, working for a common goal.
There in this verse there is a set of instructions:
1 – The delivery vehicle does not necessarily have to be a hand.
It is very likely that when you try this yourself, there is nothing springy or explosive about the body mechanics. The springiness is something that comes form relaxed (as in not tense or rigid) mind and body. To get to the point where the strength flows freely through your body and is being released from your extremities, takes a whole lot of practice and there is no known shortcut there that would help anyone get there faster...
Edit: When I say that strength flows through the body, I mean quite literally that the strength (a physical force), that was generated in your feet and directed through the waist flows freely through the body and exits through the hand that delivers it (or a foot if you are kicking).
It is definitely not Qi, I am talking about (although effectiveness of fajin may be greatly augmented by use of qi)
I can see how this explanation may sound quite mystical on the surface, but really, this is just a set of physical actions that together form a desired result. Anyone can learn to do this and in fact, many martial artists (including western arts) do that even if they do not call it fajin.
In the context of taijiquan it means power released suddenly from a correctly positioned body in the correct state of relaxation (song). The problem is that each of these steps above has to be explained. I'd rather point you at a blog post by Mike Sigman on fa jing as there are back-references there to other material. A common first step once you have developed good posture and stance is to train in "bounce jing" where you drive your rear heel into the ground and "bounce" the force back out to your hand without tensing up.