The definition of inertia is "Inertia is the resistance offered by the body whenever its state of motion is changed."
What is inertia of a body?
Is inertia actually a force exerted by the body?
If so, then why is there no application of inertia in numerical problems involving application of force on a body?
What is the actual application of inertia in theory?
There can be two answers. The first one, concerns Newton's third law. For me to extert a force upon a body and change its state of motion, the body needs to exert a force equal in magnitude and opposite in direction to me. Or, in other words, the body excerts a resistance to me. This action and reaction forces are used all the time when you consider systems if more than one particle (for example gravitational two body systems or two blocks place one on top of the other with a friction coefficient).
The other way inertia is evident in the theory is in the case of inertial forces. This are forces acting on a body, which have no appearent cause (you can't explain them from the interaction of the body with others), and they are in fact not real forces. This are forces you've got to add by "brute force" to use Newton's Second Law whenever you're not in an inertial fram of reference. The clearest example is when you are sitting on a car making a turn, you feel a pull driving you outwards. If you tried to explain that pull you wouldn't be able since this is not a real force. This is the force you feel from your body resisting to the force that acts upon you to make you change your state of motion changing the direction of your velocity vector.
I hope this was useful!
With the term "inertia" it is understood the capability of a body of opposing resistence to changes of its status of motion, caused by some forces. Remember that, in general, the motion of a mechanical system can be decomposed in center of mass (CoM) motion and a rotating motion about the CoM. Indeed, one usually speaks about the inertial mass for the first one and the inertial momentum for the latter. These quantify how strongly applications of forces (and eventually momenta) can affect the motion of the system. Physical laws expressing this link are the Newton's Second Law and the so-called "Cardinal equations of dynamics". It should also be noted that by Newton's Third Law, when you apply a force to a body, it applies the same force to you. This takes into account also the deformation about contact surfaces, so a particulary massive "hard" (i.e., only slightly deformed) body can respond in a relevant way: try to push a wall with your hand! Who can be considered the "active" agent? In such case, this is a matter of convention, although its unusual to speak about the wall as the agent. This is because there is no rotation. In fact, the physics described by a rotating observer it's different from that described on a rotating system. For example, hurricanes are due to the rotating motion of atmosphere and oceans and there is no way to introduce them by means of a rotating observer. So in this case inertia "produces" the physical effect. However, it's more appropriate to say that inertia it's a "measure" of how strong the effect is.
To be more precise, the situation is actually a bit more involved, in principle: you have to measure indipendently the force applied and the resultant acceleration and verify that they behave like collinear vectors, hence their moduli are proportional by a constant that now you can name inertial mass. Moreover, inertial mass and gravitational mass are in principle different concepts, so your experiment must be done on a surface on which the gravitational potential is constant. It's an oustanding experimental fact, coded in the strong equivalence principle, that the their numerical values coincide within the greatest precision. This turns to be essential when the system is in a gravitational field. A similar discussion holds for the inertial momentum.
In classical mechanics, all material points are endowed with some inertial mass. Massless particles are "hidden", i.e., their motion can't be described by laws of newtonian mechanics. This is deeply rooted into the geometrical structure carried by Newtonian space-time, due to the request that Galilei transformations must be the group of invariance of the theory, i.e. essentialy to the absolute character of time. In fact, in special relativity there is no absolute time and the geometric structure of spacetime allows for the description of massless particles.
In classical mechanics, inertia=force applied/acceleration applied. It's how much acceleration you apply to a body for the amount of force you apply. You could also phrase it as how much resistance a body has to change in velocity.
Inertia is not a force.
There are four fundamental forces: (1) Strong nuclear force carried by gluons, (2) Electro-magnetic force, (3) Weak force carried by intermediate vector bosons, and (4) Gravity.
In addition to the four fundamental forces, the word "force" is used to describe these mechanical operations: (1) Applied force actively exerted, (2) Normal force between stationary objects, (3) Friction, (4) Air resistance, (5) Tension, and (6) Spring force.
Inertia is determined solely by an object's mass. Inertia is an attribute of a massive object. It may be thought of as a threshold which must be overcome by any force applied to the object in order to change the object's state of motion.
Inertia has a major role in physical theory. It is the primary objective of Newton's first law of motion (http://www.physicsclassroom.com/class/newtlaws/Lesson-1/Inertia-and-Mass). In addition, inertia plays a major role in the conservation of momentum.
Inertia is the resistance of movement when a body is a subject for some sort of stress. In Newtonian physics, we can define inertia according to Newton's Second Law of Motion $\mathbf{F}=\dfrac{d\mathbf{p}}{dt}\simeq \dfrac{\Delta \mathbf{p}}{\Delta t} = m \dfrac{\Delta \mathbf{v}}{\Delta t} = m\mathbf{a}$. In this case, $m$ is the inertial mass and the mass can be said to process "a tendency of maintaining its momentum." It just means that two bodies with different masses, $m_1$ and $m_2$, will change their momentum at different rates when experiencing the same net force acting upon them.
Really, all the term "inertia" means is that a body's momentum is constant unless a net force is acting upon it. It's more or less the conclusion that objects with mass have a propriety of being "sluggish" when a force is acting on it; it will not change its momentum instantly.
There are a lot of complex answers. I am just gonna give you a simple one. When we talk about Inertia, just think of it as a property of a body that allows it to stay in its state of rest or uniform motion. Just think of it as a resistance to change in the state of body. Just for a simple example, say you have a pitcher of water, if you put your finger in and rotate in the anti-clockwise direction, the start presents a little resistance. Don't mistake it for fluid resistance, it is just an example. Then if you suddenly try to change the direction to clock-wise, the water in motion with your finger experiences some resistance, and that resistance is the definition of Inertia. So, since Inertia is just a property of a body, we don't consider it force. Actually, it a opposition to the difference in force. So, the actual applications of Inertia is just say to make a body continue in it's state, even after experiencing an opposite force.
Inertia is considered in physics when we want to do work in sense of physics. It's not a force exerted by body instead it is force need by body to move. If it was a force exerted by body then equation will itself be changed. In equation acceleration is the need one to move it. If it was exerted by body then mass and acceleration would not be mentioned it would have been said that it is it's insintric property. Well it's not because it can be measured other wise it would be a fundamental quantity and not a derived one
