Eidos is an ancient philosophical term; loosely speaking, it means the information that makes up an object we then perceive through the material world. When we see something – for example a chair – we see the manifestations of it’s properties – we see wood, four legs, we can even estimate weight, height, velocity, and thousands of other conceptual variables we cannot see. Eidos is where all those variables are defined. The material manifestation directly effects Eidos – if you kick a chair, you suddenly add directional acceleration to it, and you may even – depending on it’s properties of durability – break it into different objects.
The opposite – the altering of somethings properties in Eidos effecting the material form – as it turns out, is also true. This is what we call Magic.
The Eidos key is a theoretical concept by people that study how magic works, and is the so called "missing step" between been calculation and actualization. To use magic, a mage must first conceptualize the change, then calculate it, and then actualize it into the world. Many researchers delve into how a mage takes a calculation applies it to the world, but many leading theorists support the idea of a so called "Eidos Key" unique to humans that allows them to interface directly with Eidos.
Several attempts have been made to make an artificial Eidos Key to make a computer fully capable of independently activating magic, but so far no one seems to have succeeded. Well, unless you count I.L.A., but that is a little controversial currently.
Many mages refer to Form of something in Eidos as Eidos data; this is a construction of the mage's mind to make processing the calculations of Eidos forms possible. Eidos data for an object contains all of the essential properties of that Form, as well as all of the conditional properties. It is possible to imagine this as a multi tiered structure, with the conditional properties being more accessible than the essential properties. Essential properties are very challenging to read or understand; at their most basic level, they contain things like material composition; things that if changed, the Form would no longer be the same thing in the material world as it currently is. Conditional properties are the things that tend to change on a regular basis, or that do not fundamentally change what it is - some of these are obvious, such as current relative movement speed to other objects, while some have more complicated ramifications, like weight. While weight should be a derived property from the other data of the form, that is a result of physics, and it is possible to overwrite the data directly in Eidos, though such a change will revert as soon as the mage that made the change is no longer actively persisting the calculation as physics rewrites the correctly derived value.
How much a mage can actually "see" Eidos data varies drastically. For many mages, they can see very little Eidos data, and have to rely on preconceived calculations to make changes. You do not necessarily need to see an objects data to change the objects speed as long as you have what that calculation would be memorized or programmed into an auto caster. More rarely there are people that read a quite a bit from Eidos data, but this tends to be extremely taxing as even limited Eidos data is fundamentally more data than a human brain can easily process, and it tends cause strain to parse relevant information from the data stream - like trying to read a from a screen that flashing different images a dozen times a second.
While the use of magic can (and does) violate the conservation of energy, there is a different sort of energy required to initiate changes directly to Eidos; the most wildly adopted term for that energy is mana, though it is also know as Psonic particles or psonic energy, particularly by Central.
The amount of mana required to actualize a change is not equal to the amount of energy being created by change in the physical world, but how much the change to Eidos violates somethings fundamental properties. These, however, frequently overlap to some degree.