Here are two modern definitions of inertia:
Wikipedia https://en.wikipedia.org/wiki/Inertia: "Inertia is the resistance of any physical object to any change in its state of motion."
Merriam-Webster dictionary https://www.merriam-webster.com/dictionary/inertia: "A property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force"
Notice that in both definitions inertia is a property of an object or matter! This is very misleading, because it doesn't reflect the physics (even at a high level) of how inertia is applied to matter. Inertia is an inherent property of space, it's not a property of matter, although it applies to matter! It is proportional to the amount of matter in an object (mass), no doubt about that, but it is really a force of resistance that a space applies to an accelerating/decelerating object. A definition of inertia along these lines would provide much better picture of a real situation, real physical process.
Because of our current definition of inertia, a lot of people have a lack of understanding of basic things:
Why does kinetic energy increase quadratically, not linearly, with speed? Why does kinetic energy increase quadratically, not linearly, with speed?
Why is the amount of work required to accelerate a body from 10m/s to 20m/s three times the work needed to accelerate a body from 0m/s to 10m/s?https://www.reddit.com/r/askscience/comments/40bi9z/why_is_the_amount_of_work_required_to_accelerate/
Notice that these are very popular questions, but no one gave a satisfactory answer to these questions, as authors of these questions complain. People want to know the real physical reason; they want to understand WHY kinetic energy is speed square, WHY it makes sense to define work as force by distance. Answers like "the work was just DEFINED like that and the rest follows" - these answers do not answer the actual questions people ask.
So INERTIA is the real answer to these questions! If you just make it clear to people that every bit, every meter of space resists acceleration, then it's quite obvious why energy increases quadratically with speed. Inertia is much like friction with the space, the only difference is - this friction only applies when you are changing your velocity; it does not apply at constant velocity. So the more space you travel with acceleation, the more friction you need to overcome, the more work you have to do (evergy to spend) to overcome that friction (force of resistance to your acceleration). So that's why the energy you spend to maintain acceleation is proportional to distance you travel with that acceleation. And, when you go with constant acceleration, the distance traveled is proportional to speed square.
Inertia is essentially a force applied on an object that moves with acceleation. So when the object accelerates in space, there are two forces acting on it - the force that pushes it and the force of inertia that resists the acceleration. By second law of Newton, the resulting acceleration is proportional to the resulting force, which is pushing force minus force of inertia. Let's illustrate it using the question from reddit.com: When you accelerate a body from 0m/s to 10m/s, you travel 5 meters in first second. Same acceleration continues, and during the next second a body changes speed from 10m/s to 20m/s, traveling 15 meters. So - same acceleration for the same amount of time (same resulting force), but a body traveled 3 times more distance. 3 times more resistance to overcome -> 3 times more energy to spend. So during the next second you have to apply more pushing force on your object, to maintain the same aceleration, than during the first second. Simply because the resistance is higher.
Should we really consider inertia as a real natural force that applies on matter during space-matter interaction under acceleration? I am not sure, but it is certain that if we look at it that way, the above questions become simple to answer. As to WHY and HOW inertia acts like this, what is the real mechanism - nobody knows so far (2017). There are theories but all of them have issues.
The idea that inertia is a property of space, is not new. Albert Einstein tried to modify it, as in GR objects modify space, so inertia becomes a property of object-space interaction. More recently, there were several research efforts trying to explain inertia as a force acting in space against accelerating matter:
http://www.calphysics.org/haisch/science.html ; https://physics.fullerton.edu/~jimw/general/inertia/
Inertia seems to be one of the most fundamental properties of space and our world. It is as fundamental as energy, if not more. Stable world is not possible without inertia. If there is no inertia, a simple push would accelerate an object to infinite speeds (let's forget relativity here), and that same object could be completely and immediately stopped after collision with a small piece of space dust. That object wouldn't carry any energy. Even the famous E = mc2 shows that no inertia results in no energia. Space must have inertia in order for energy to exist in forms we know (concentrated as physical matter in fixed locations in space). To put it philosophically, Vishnu (inertia) keeps the world stable while Shiva (energia) tries to tear it apart.