Newton's first law: is his concept of (force of ) inertia still useful and used? 
The force of inertia is the property common to all bodies that remain
  in their state, either at rest or in motion, unless some external
  cause is introduced to make them alter this state.

That is the definition by Jean d'Alembert in the Encyclopedia (1757) he explains (this is lost in the bad translation from French) that he used 'property' and not 'power' because he believes that this word evokes a metaphysical being resident in the body. (sic: sort of poltergeist). And this is the original wording by Newton: (1726), in definition III

*Materiae vis insita [internal/resident/lit: implanted force] est potentia resistendi, qua corpus unumquodque, quantum in se est, perseverat in statu suo vel quiescendi
  vel movendi uniformiter in directum.*

The introduction in italics [the internal force of matter is the power to resist..] is prodromic to the first law (see below), but what is interesting is the explanation thereof given:

Haec semper proportionalis est suo corpori, neque differt quicquam ab
  inertia massae, nisi in modo concipiendi. Per inertiam materiae fit, ut corpus omne de statu suo vel quiescendi vel movendi
  difficulter deturbetur. Unde etiam vis insita nomine
  significantissimo
  vis inertiae dici possit.

This [internal force] is always proportional to its body and is not different in any way from the inertia of its mass but for [our] way of conceiving it. It is because of the inertia of matter that it is more difficult to alter the state of a body of being at rest or in motion. Therefore this internal force may also by called by the most significant name of force of inertia.

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi
  uniformiter in directum, nisi quatenus a viribus impressis cogitur
  statum illum mutare.
Law I: Every body persists in its state of being at rest or of moving
  uniformly in a straight line unless it is compelled to
  change its state by forces impressed. 

[motive] force impressed vi motrici impressae is also the denomination of the esternal force that produces [kinetic] energy in the second law.
As you can see from the original texts (by Newton and d'Alembert), there is no difference whatsoever between 'inertia' [= sloth] and 'force of inertia' this latter denomination is a sort of 'promotion' to a higher status of conception: the 'laziness' of mass/matter is (proportional to) mass and is the 'power' that opposes change, it is an internal 'force': the 'force of laziness/ inactivity' of all matter. In the original post I quoted verbatim this latter definition from the  Principia and pointed out the amusing oxymoron, this was taken as 'hostile' [sic], 'unclear' by hasty, superficial or poorly-informed readers andas 'nonsensical babble'[sic] by Olin Lathrop.
All the arguments to criticize this question, to condemn it as 'unclear', to close it, eventually delete it altogether and differ its reopening seem rather unjustified hair-splitting and (themselves really) incomprehensible. 
I am striving to reopen it as a matter of principle, because a distinguished member (Rod Vance) has repeatedly stated in comments, alas deleted, that he has a very interesting answer. I also want an opportunity to learn what exactly is my 'nonsensical babble' and to learn how to make this question 'clear' and comprehensible, hoping the worthy members who inexplicably ostracized this question (and btw the downvoters) to explain what is wrong with this question:


*

*Is the concept of [force of] inertia still useful and used? 

*Is it now just one of the fictitious forces or what?

*Can you list a few situations in which, if we didn't use this tool we might be in difficulty?


It is possible that the two terms have acquired, in use, different meanings of which I am not aware, some might (wrongly) assume that moment of inertia is a different 'concept' from 'force of inertia', or other:

Are you asking about inertia in general, or just the term force of
  inertia? Please edit the question and title accordingly. Using
  parenthesis (force of) inertia is ambivalent. –  Qmechanic
Is this a semantic/linguistic question about the term force of
  inertia (as opposed to e.g. the terms
  fictitious force, pseudo force, or inertial force)? –  Qmechanic

I suppose that after these authoritative comments a fourth additional question is necessary:


*

*What is the (physics) difference between: 'inertia', 'force of inertia' and 'inertial force'?


It would be of great interest for everybody, I suppose, to learn when and how the to terms diverged and if they have a different fate. I left the parenthesis because I am enquiring about both terms.
 A: The concept of inertia is indeed useful in two ways. I think your notion of it as a technical promotion of the everyday word "sloth" (without the baggage given it by the Roman Catholic translation of the "deadly sin" Ἀκηδία) as extremely close to the mark. In physics the notion of "inertia" has two, very alike uses:

*

*The first is practical, through a weak form of D'Alembert's principle. The notion arises where we look at a system from an accelerated frame of reference and treat it as a non-accelerated one: to keep the things making up the system "together" and "still" relative to the accelerated frame, we imagine that each of the system components are exerting a force of inertia (in the sense exactly described by Newton in your quote) on the system that tries to "tear it away" from the frame of reference wherein our discourse takes place. This "force" arises from each component's "sloth", i.e. resistance to any change of their state of motion from a non accelerated frame (let's leave aside the term "inertial frame" for this latter notion for now). There has to be something tethering each of the system components to the accelerated frame to resist the "force of inertia" that each of the components exerts in "trying to tear away" from the frame and resume a uniform state of motion. Thus, in designing a centrifugal pump in this way, we would imagine the impellor sitting still, but each of the blades exerts its centrifugal force on the impellor's hub and we thus see that the hub and blades are in a state of tension to resist this centrifugal force and must accordingly be designed so that they may be strong enough to yield this resistance. From a non-accelerated frame, we would simply see the blades making circular paths, and thus we conclude that they are accelerating, so, by Newton II, we know that the hub must be pulling the blades radially, i.e. providing the centripetal force needed to set up this accelerated motion. Sometimes D'Alembert's Principle is thought of simply a re-arrangement of Newton's second law, and thus derivable from the latter, but this is not so as discussed in QMechanic's answer here to the Physics SE question "Deriving D'Alembert's Principle". Moreover, it is indispensable in rotating rigid body problems. In this kind of problem, if we try to work only in non-accelerated frames, Euler's Second Law of Motion becomes highly awkward, because the inertia tensor $I$ of a spinning body is constantly changing relative to a non-rotating frame. It is much easier to fix our frame to the spinning body, thus benefitting from a constant inertia tensor $I$ and live with the inertial forces $\omega\times(I\,\omega)$ in the Euler equations $M = I\, {\rm d}_t \omega + \omega\times(I\,\omega)$


*The great theoretical utility of the notion of inertia is as inertial mass: this notion is useful simply by differing from the notion of gravitational mass. Without a clear understanding of the stark differences between these two notions, there could be no discussion of the Equivalence Principle (see Wikipedia page of this name). In this form, the notion of inertia is an indispensable part of the epistemology of the General Theory of Relativity, so I shall now concentrate on talking about this use of the "inertia" notion.
Inertia, Gravitational Coupling and The Equivalence Principle
So now we look at the meaning of the word "mass": it actually has two (and possibly three) in principle distinct meanings:

*

*As "inertia" or "inertial mass" it is a measure of the body's "sloth" or "shove resistance" as discussed above, i.e. inversely proportional to its acceleration under a unit imbalanced force (i.e. inversely proportional to the body's "response" to a standard force). Thus this notion is expressed by the quantity $m_I$ in Newton's second law $\vec{F} = m_I \,\vec{a}$;


*As a "coupling constant" describing how strongly a body is influenced by a gravitational field, i.e. how much nett force a unit gravitational field imparts to a body (in the Newtonian notion of gravity). Thus this notion is expressed by the quantity $m_g$ when we say that a small test mass of gravitational mass $m_g$ in a gravitational field $\vec{g}$ feels a force $m_g\,\vec{g}$;


*A third possible notion, not really important here, is as a measure of a particle's "stay puttability" as I discuss in my answer to the Physics SE question "Can mass be directly measured without measuring its weight?". This is simply how still a body of a given, standard degree of localisation in space can be made and still conform with the Heisenberg Uncertainty Principle.
Ponder the first two carefully and take heed how much in principle different they are as notions. Without further information, experimental results or postulates, I hope you will agree that there is no possible way whereby these two notions could a priori proven or even guessed to be the same.
The weak equivalence principle asserts that notions 1) and 2) above are the same and, for any body, regardless of its makeup or quantum state, we have $m_I = m_g$ (see my footnote 3). These two are not just alike. They are exactly the same. This is a stunning assertion and it still dazzles me to this day, even though I am fifty years old and first read about it when I was fourteen (I did, however, take a further six years to fully appreciate its significance).
Given their vast conceptual difference as physical notions, any assertion that they are the same must encode real, falsifiable physics about gravitation. For what it means is that any small mass, regardless of makeup or quantum state, with a given initial velocity in a gravitational field must undergo exactly the same motion. This principle has been clearly recognised by many scientists for nearly fifteen hundred years. In  the sixth century CE, John Philoponus (see Wiki page of same name) said of experiments potentially falsifying the equivalence principle:

"But this [view of Aristotle that the time taken for a body to fall a given distance is inversely proportional to its weight] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights, one many times heavier than the other you will see that the ratio of the times required for the motion does not depend [solely] on the weights, but that the difference in time is very small."

Galileo certainly knew the equivalence principle and his famous experiment dropping different weight balls from the Tower of Pisa was almost certainly, in reality, done in about 1586 by Simon Stevin dropping balls from the Delft churchtower (see discussion in Equivalence Principle Wiki Page).
Many careful experiments probing the equivalence principle's correctness have been done; amongst the most famous are the Eötvös Experiment (see Wikipedia page of same name) as well as those of Newton (in finding that pendulums of the same length have the same period, independent of length) and by the commander of Apollo 15, David Scott, when he dropped a feather and a hammer from the same height on the Moon to see them both hit the ground at exactly the same time.
So now on to General Theory of Relativity. Einstein was convinced that the Equivalence Principle would lead him to his GTR and from very early on in the piece he kept returning to this principle. The principle shows itself very blatantly in his early works before the full GTR paper of 1916. In:
A. Einstein, "Über den Einfluss der Schwerkraft auf die Ausbreiitung des Lichtes", Annalen der Physik, 35, 1911 English version "On the Influence of Gravitation on the Propagation of Light" is here)
he uses nothing but the equivalence principle very directly and on its own to derive, by very simple and clear arguments, some of the important, readily falsifiable results that would follow from his later 1916 paper:
A. Einstein, "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49, 1916 (English translation, "The Foundation of the General Theory of Relativity" is here)
In the latter paper, indeed in the teaching of GTR, the equivalence principle seems to slip into the background a little (a great deal, in some modern texts) and often the direct assertion of the equivalence of mass notions is overshadowed in modern texts by something like the following statement:
The tangent space to the spacetime manifold solution to the Einstein field equations is Minkowskian
or
Spacetime is locally Minkowskian
or something like this. This is indeed a reasonable, indeed a stronger, statement of the equivalence principle, but it does, in my opinion, need some further explanation. It's at first glance quite different from the Einstein version of the equivalence principle:

The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime

The way that the equivalence principle is encoded is, in my opinion, some of the reason why inertia is not discussed very much in relativity. The EP is actually part of the building materials for the GTR: the very assertion that spacetime under the influence of "matter" (anything with energy content and thus gravitational mass) is a differentiable, indeed, pseudo-Riemannian manifold fully encodes the EP. So, the mere choice of the geometrical object, before we even contemplate writing down the Einstein Field Equations, or indeed the physics behind them, fully encodes the EP. A manifold is locally like a Euclidean (or, in GTR, flat Minkowskian) space: there are other geometrical objects, notably an Algebraic Variety that we might have chosen to describe the "curvature" of spacetime with and which are more general than manifolds and which do NOT encode the EP. To examine the manifold and why it encodes the EP, I'm going to give my version of the Einstein equivalence principle:
For any chosen, positive precision $\epsilon>0$, there is a magnification $M$ such that if you look at the spacetime manifold with this high enough magnification, you will see a laboratory indistinguishable (to within the chosen precision) from the main cabin of Salviati's Ship
Salviati's ship (see the Wikipedia page for "Galileo's Ship") was, of course, a thought experiment wherein Galileo asserts the impossibility of telling whether or not a ship is moving uniformly by any experiment which does not look to an outside reference. The mere fact that the spacetime manifold has a Tangent Space at every point alone means that if we zoom into the manifold enough so that our little ship takes up a small enough volume in space time, then there exists a frame of reference, that moving along a geodetic line, such that if the ship were stationary in that frame of reference, the Salviati thought experiment would hold. It might (inside a highly curved region like near a black hole) have to be a volume that it is comparable in size to an atomic nucleus, but that's OK in GTR: GTR is a classical theory that does not see the World's granularity: in principle there is always a Salviati ship, even if it is the size of a proton and exists only for $10^{-20}$ seconds. This is a freefall frame, the most general concept of an inertial frame, or a "slothful frame" that describes a (geodesic) flow on spacetime arising in the absence of external forces, and that any accelerated motion relative to that frame requires an unbalanced force. It describes how something moving through spacetime "wants to move" and has a "stubbornness to move thus" and must be compelled by unbalanced force to move differently.
So suppose we are on Salviati's ship, freefalling in a uniform gravitational field and the equivalence principle did not hold, and no frame of reference were Minkowskian (an inertial frame, in Special Relativity). The butterflies, being of a different makeup, might accelerate differently from the waterdrops from the bottle, and the ship's cat, being of very different makeup, would have accelerated relative to the scene and been lost long ago! The whole scene can only stay fixed, with all its constituents staying at the same relative positions, if the equivalence principle holds. Thus we see that the differentiable manifold conception of spacetime can hold only if the equivalence principle is true.
So there is always a local "inertial frame" (I actually like the word "freefall frame" better) in General Relativity. We have come the full circle: for now how do we describe your frame of reference as you sit "stationary" on the Earth's surface reading this? Think about your bottom: you feel it is being pressed by your seat. You conclude that there must be a force pressing you into the seat: our mother tongue has a word for this force: your weight. But this is an inertial force in the sense described at the very beginning of my answer. For it so happens that, in the presence of the Earth, the true inertial frame, the true cabin of the Salviati Ship, as described by GTR, is one beginning stationary relative to you but which is "accelerating" relative to you at $g {\rm m\,s^{-2}}$ towards the centre of the Earth. The matter of your seat must therefore push against you with force $m\,g$ upwards to beget your acceleration relative to the Salviati ship. But, of course, we find it easiest to think in a frame of reference that is stationary relative to our Earthly home. So, in this accelerated frame, we feel the inertia of our bodies as they "try to tear away" and follow their natural, inertial frames.


*

*See Eduardo Guerras Valera's wonderful answer to "How (or why) equivalence principle led to Einstein field equations?" for a fuller description of how Einstein seems to have embedded the EP into the GTR - the modern manifold concept I described was not how people thought about manifolds in Einstein's day, when they thought of them as needfully being curved objects in a higher dimensional Euclidean space. The two conceptions were only shown to be equivalent notions in the 1940s by Hassler Whitney and 1950s by John Nash (the mathematician depicted by Russell Crowe in the film "A Beautiful Mind").


*Some theorists believe that the EP is so NONtrivial that its very breakdown (actual experimental falsification) may be the first place where we see in practice the General Theory of Relativity yield to more general, yet to be developed quantum theory of gravity. See the discussion in vnb's answer to the Physics SE question "Does quantum mechanics violate the equivalence principle?". Indeed Paul Davies in his article "Quantum mechanics and the equivalence principle" shows a possible chink: for quantum particles tunnelling to regions in a gravitational field whence they are classical forbidden, the tunnelling depth depends on the particle mass. Also, the problem of whether an electrical charge on the Earth's surface radiates does not seem to be fully resolved. See Ben Crowell's answer to the Physics SE question "Does a constantly accelerating charged particle emit em radiation or not?".


*Strictly speaking, all the physics of the Equivalence Principle would be encoded by an assertion that $m_I = \lambda\,m_g$, where $\lambda$ is any constant, so $m_I=m_g$ encodes the EP together with a choice of scaling constant. We choose to define the universal gravitation constant $G$ in Newton's law of gravitation so that $\lambda = 1$. Actually, in General Relativity, in some ways it would make more sense to define $G/(8\pi)$ to be the gravitation constant, in which case the equivalence principle would be  $m_I = \sqrt{8\,\pi}\,m_g$.
A: The question of mass has arguably been one of the two most important issues in physics (the other being the electromagnetism). Physics has tried to uncover the true nature of mass for hundreds of years, to no avail so far. Not surprisingly, its description is somewhat circular: 
“In physics, mass is a property of a physical body which determines the body's resistance to being accelerated by a force and the strength of its mutual gravitational attraction with other bodies.”
where „the body's resistance to being accelerated” is, obviously, inertia, which is „one of the primary manifestations of mass”, while gravity „is the only force acting on all particles with mass”.
To answer bobby's question: the most successful (the only?) theory that has done away with inertia is the General Theory of Relativity. Einstein aimed at extending the Relativity Theory to gravitation, and while working on this problem, he found inertia to be the major obstacle, for rather obvious reasons. The equations of SR could not be used for non-inertial frames of reference, and gravity is all about acceleration. In his letter to Born, Einstein said: "the gravitational equations would still be convincing, because they avoid the inertial system (the phantom which affects everything but which is not itself affected)." Einstein was able to claim this, because he made two truly amazing insights: being at rest on the surface of Earth is equivalent to being in an elevator accelerating up, and also - perhaps more importantly in the context of inertia itself - when freefalling toward the source of gravitation, one feels no acceleration at all. As Einstein was looking for the equations for motion, and the movement under gravity turned out to show no proper acceleration (the acceleration is only coordinate), he was allowed to assign an inertial frame of reference to this movement. Consequently, gravity was called a fictitious force. Problem solved. 
Well, was it? Aside from the fact that acceleration is still measurable on the surface of the source of gravity, there is also another question left. Apparently, Einstein missed one thing: the geometrical space-time curvature concept, which replaced the force of gravity in GR, quite well explains the movement along geodesics, but does not explain the very impulse to motion in this field. The curvature by itself cannot make things move. If there is no force “underneath” the curvature, there is no reason the body should move at all. The usual counter-argument at this moment is that this is not a problem in GR, because under the concept of space-time and velocity 4-vector, each body is always in motion along the time axis. Assuming time is really orthogonal to space, Newton's first law of motion still says that one needs a force not only to set a body in motion, but also to change the direction of its motion. A body that moves along the $t$ axis requires a force to change the direction of its motion toward any of the space axes. And this change without a force is unexplained. This omission is a really serious one, since the ability to make things move is by far the most important aspect of mass and gravity.
Now, all these considerations make the concept of force of gravity – or to be precise, acceleration – is still valid. This means that the force of inertia is also still in play. OK, so what is inertia then? 
„Inertia is the resistance of any physical object to any change in its state of motion, including changes to its speed and direction.”
Yeah, we all know it. What we do not know, however, is how this "resistance to any change in motion" is produced.
Let's follow a simple line of logic then. What is a change in motion? Acceleration obviously. So inertia resists any acceleration. Now, what does it take to resist acceleration? Another acceleration. Would that suggest that inertia is acceleration itself? Well, each massive body generates gravitational field, and gravitation is acceleration. All seems to fit in.
How would that work in practice? If one is trying to move a material body, one must work against the body's own acceleration pointing outwards. It is a fact as long as Einstein's equivalence principle holds true. The more massive the body, the bigger the acceleration it produces, and the more external force it takes to work against this acceleration.
Seems very simple, although not intuitive. But hey! If there is a single most often repeated statement in contemporary physics, it would be probably this one: “Intuition is not the final argument in science; science is about models, equations and predictions”. True. Inertial mass has been proven to be equal to gravitational mass, and therefore the force required to move inertial mass must exceed its force of gravity at the surface.

To sum up, addressing the original questions by bobby:

1) What is the (physics) difference between: 'inertia', 'force of inertia' and 'inertial force'?

There is none. All these terms express the same property of mass - its innate resistance to external force (acceleration). What makes inertia special is that it resists acceleration (force), and it takes another acceleration (force) to do that.
Also, following the line of reasoning above, which is confirmed by the famous Eötvös experiment, we can say that there is yet another synonym to inertia - gravitation. And following the Ockham's razor principle, it would only be logical to assume twin properties - inertia and gravity - to be simply one and the same thing.

2) Is it now just one of the fictitious forces or what?

If understood correctly, no. Inertia is the fundamental reason why it requires a real force to change the motion of a body. It's real, because mass, and nothing else, really resists a change to its motion. Also, inertia, being a synonym of gravity, is - as shown by Einstein - a real acceleration when measured at the surface of the body (source). 

3) Is the concept of [force of] inertia still useful and used?
4) Can you list a few situations in which, if we didn't use this tool we might be in difficulty?

I take these two questions as provocative, or intending to explicitly demonstrate that dismissing (force of) inertia is not so wise an idea, to say the least ...
Whenever there is mass influencing the mechanics and equations of motion, there is always the concept of inertia involved. Because mass could (and should) be understood just as another synonym for inertia.
As to the situations where inertia - and therefore mass - cannot be neglected as a concept. There were some examples given in the comments (by Jim) to the question. There is a plenitude of examples: road traffic (car safety belts, bumpers, road railings, safety helmets), planes, lifts/elevators, or even building designs. So, in all situations, where mass (understood as acceleration out and therefore exerting real force on other objects it is in contact with) affects the equations and reality they describe, inertia is included by definition, and therefore dismissing it would get us into trouble. Even relativity that is believed to do just fine without (force of) inertia does provide equation for relativistic mass. Why? Because inertia is a fundamental factor when dealing with motion, and even particle physics must accept that.
Winding it all up - unless we manage to get rid of the concept of mass, we also cannot get rid of its synonym, inertia (or another synonym, gravity) - as simple as that.
A: Other answers being good, i'll add 2 more cents here.
Inertia as quoted in question (btw, this is nice, to see actual historical sources), is (defined as) the tendency of matter to continue in its current state, unless changed.
This is inertia, the term "force of inertia" is just another way to state the same thing but using Newton's second law, $F=ma$, so this $F$ force, needed to change a state of a system, must be equal or overcome its inertia, or in force terms, the force of inertia (i would say just a manner of speaking).
Elaborating a little on the previous statements:
Inertia is associated with Newton's 1st Law (i.e "the tendency of matter to continue in its current state, unless changed"), but one could transpose this concept into Newton's 2nd Law $F=ma$, and talk (i would say just a manner of speaking) about a force of inertia, in the sense of the effects of the inertia concept when associated with the 2nd Law. Literally, Newton (and others), did not define a force of inertia (as for example a force of gravity), so this phrase is an analogy in the context of forces associated with the 2nd Law. Again i would say it is just a manner of speaking about inertia either in the context of the 1st or in the context of the 2nd Law (which involves and defines the concept of force). Two facets of the same thing.
The connection of inertia to mass is also correct in the sense (as the 2nd Law makes explicit), that in order for an object to acquire a given change of velocity (i.e acceleration) different amount of force would be needed depending on its mass. 
i would say (see below) that "inertia is not one of the primary manifestations of mass" (as in the wkipedia article) but the other way around (i.e "mass is one of the manifestations of inertia"), as inertia is more general concept (which in the Newtonian framework is associated with mass). 
For the previous example, for the same mass object, a different amount of force would again be needed for it to acquire a certain given change of velocity based on its previous velocity (isn't this a manifestation of inertia as well?). Furthermore there are massless particles which can have inertia, in the sense stated on top (which is associated with other properties).
Is inertia useful? Well the correct question is "is inertia physicaly relevant?"
Yes. (The detailed answer would have to venture into many fields, but i'll try to give a small summary)
Inertia is analogous, in a sense, to relativity, in the same way that a signal cannot travel instantaneously, the same way inertia works on system changes.
Inertia (in another perspective) describes collectively the correlations of a given system with the rest of the environment (here we can enter thermodynamics description).
Inertia is a part of the clasical mechanics framework (there are s few others) which provide connections to the other important physical fields (like thermodynamics).
Furthermore the previous paragraphs relate inertia to (at least some part of) causality (another cause needed to change the state of a system that resulted from some other previous cause).
Consider a converse question. "How would things interact without inertia?"
In still another sense, inertia is the manifestation of the negative feedback loop concept (a-la cybernetics) which provide dynamic stability to systems (again related to thermodynamics point of view mentioned earlier). 
A small note on inertial/fictitious forces. Inertial forces are not fictitious. Each and every observer will actually feel them. They are non-assigned forces For example, in the sense of electric forces (assigned to electric charge of a specific particle) or gravitational forces.
They are non-assigned in this sense, however they are assigned to the relative motion between systems of objects.
This is a huge (un-investigated to a large extend) topic, so i'll just leave it here.
