Once again, I apologize for the late update. I will make it longer than I had planned to. Anyway, on to the story.
Eidolon is the term for any spirit that is not a live human or Nephilim. While the term initially applied only to sentient beings, years of use as a catchall term for anything with a fixed powerset has made this the common meaning.
Eidolon are almost always created partially from materials in the environment they appear in, and natural eidolon are either born or created by a combination of the expectations and beliefs of humans and Nephilim, with the level of effect being based on proximity.
All timelines excepting parts of the Fourth have had magic for long enough that any area with a significant concentration of Fuel will also be a spawning ground for eidolon appropriate to said area.
Most sentient eidolon can reproduce with each other, and most non-sentient eidolon can reproduce with each other if sufficiently similar.
All Eidolon capable of producing Nephilim are sentient, and very rarely have no human form.
Nephilim are the offspring of humans and eidolon, eidolon and nephilim, or nephilim and other nephilim.
They share the magical attributes of both parents, and often have a unique powerset that combines said attributes. Physical attributes default to the human side, but magical attributes can modify appearance, most commonly the eyes. While some types of nonsentient Eidolon can reproduce with similar animals, the offspring are considered new Eidolons.
Affinities are the result of a mage focusing on a specific type of effects, and make similar spells become easier to cast and require less Fuel for the same effect. Affinities are a combination of the soul naturally adapting to new ideas and the archetype that sprung up around that effect. An archetype is the difference between an Engram and a spell; a spell is any Engram established enough to have it's own archetype.
Archetype Pieces are devices, often weapons, often already magical, that have become well enough known for a specific set of attributes that they develop a powerset based on them. They are usually quite useful, but the same effect can make badly designed or merely not well liked items worse.
Incidentally, in this world, a standard burial practice is to bury the primary tool of the deceased (or a replica, if the original is unavailable), piercing the chest cavity. While this discouraged heirlooms, it did encourage new development, apprenticeship from an early age, and the creation of minor Archetype Pieces. It is thought to have originally have been a practice for soldiers, allowing their spears to double as gravestones when there was no time to carve one properly.