There's been a remarkable amount of B.S. written and put online about fa-jin. There's also been a good amount of B.S. written questioning its very existence. Here's an attempt to contribute something serious on the issue.
At left are two characters. The distinction between Jin and Jing was introduced for non Chinese-speakers to distinguish between the expression of martial force (at the top) from the lower term which refers to one of the Three Jewels of Internal Alchemy.
Jin means refined or cultivated energy or power. The large seal symbol for Jin shows strength (Li), recruitment or unification, and underground streams or paths (Jin-lu). So the implication combines strength, paths or streams, underground (hidden), and recruitment. ( I have no Chinese so I'm relying here on the translations of others.)
Dr. Shen Zaiwen said that fa-jin was to be found in any martial art. That it was typified by whole-body movement, but that it differed from one art to another. External fa-jin relied on muscularity and contraction and tended to be big and long. Internal fa-jin relied more on relaxation and tended to be shorter and more compressed. He said that the fa-jin of Xing-yi was long and powerful, that of Bagwa was spiraling and twisty, and that of Yang was small and compressed. He was an Old Yang master. Personally I find the traditional Yang of the family at present to also be reasonably long.
Of the basic Tai Chi Classics only that of Wu Yu-xiang deals explicitly with physical issues. In it are two definitions which I believe describe two categories of fa-jin.
Power coming up from the feet, released through the thighs, directed by the waist and manifesting in the hands. (Eg. Slant Flying)
Sinking into a clarity of relaxation and thinking in one direction. (Eg. short jin)
In the Yang / Wu tradition there is a taxonomy of 25 jins. It is useful but incomplete as various important jins do not make it onto the list. Fa-jin simply refers to the expression of any jin. It can be long, short, explosive, shuddering, slow, etc. It can be going out, coming in, sinking, or rising.
Most think of fa-jin as explosive; not so. But, all Tai Chi fa-jins have certain characteristics in common. Some of these are: centredness, focused intent, lack of reliance on muscular mass and contraction, powerful looseness, correct structure, etc. Usually fa-jin involves three stages: entry, progression, and completion.
Fa-jin in Tai Chi is one pons asinorum (colloquially among architects: "Bridge of Arseholes"). Developing reasonable fa-jin is a major objective of the intermediate stage of Tai Chi practice.