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.