What exacly is the so-called "formless" jhana? When reading the suttas, sometimes I come across to the expressions: 
"dimension of the infinitude of space",
"dimension of the infinitude of consciousness", 
"dimension of nothingness" and
"dimension of neither perception nor non-perception".

What exactly is the meaning of the expressions above? Why they are "formless"? 
When the Buddha describes Right Concentration, he mentions only the four "form" jhana, why he excluded the "formless" jhana as Right Concentration?
Here is my understanding as of 2018.

When they say "form" (rupa) or "seed" (bija) in context of meditation, it's what we call an object of meditation these days. For example, your breath, an image of the Buddha, a particular repulsive image used as an antidote to lust - are all examples of meditation objects. An object of meditation is used as its focal point and directly correlates with the meditation's goal or objective.

Formless meditation, in contrast, is a type of meditation that involves no particular focal point, no object. As I understand, the formless jhanas refer to a step-by-step sequence of getting rid of any notion of object, however subtle. 

But before we can talk about formless meditation we need to talk about the ancient model describing the process of perception. In that model, the way we perceive things is by identifying a mark (aka sign) that serves as the main determining characteristic of an object, and by matching that mark against our memory of previous impressions, making an associative interpretation that what we're looking at (hearing, thinking about) can be described with such-and-such concept. This initial hypothesis is then confirmed by finding other secondary marks (signs) that the identified object must possess and confirming they lead to the same association. This process repeats in cycles until one of the associations expands into a related concept or a memory, which leads down some train of thoughts with its desires, fears, and other problematic mindstates.

Since thinking was understood to be the root of all evil, and yoga was defined as "citta vrtti nirodha" (stopping the ever unfolding associative cycle), getting rid of all objects was the chief method for stopping the thoughts. This was the state of the art before Buddha.

Now that we understand what we're dealing with, we can understand the sequence of formless jhanas. First, you visualize infinite empty space with no objects. This is a good approximation of objectlessness, but if you think about it, you still have an object on your mind - the image of empty space. In fact, this image of empty space consists of some kind of mental "mark" or "sign" that the meditator assumes to be a good associative hook (e.g. a tactile sensation of moving and encountering no contact, or a spatial image of endlessly expanding in all directions etc.) plus the concept "empty space" that we are interpreting that sign into. So, upon close examination this type of meditation is not very objectless as it still involves what's these days is called semiosis or the act of interpreting a sign into a meaning.

So as a next step, you meditate on the empty mind with no objects. It's a mind that is potentially capable of perception, it could potentially have an object but it does not. At first you successfully imagine such a mind and in contrast with meditation on empty space, this does seem to involve no sign and no act of interpretation. It seems like a truly objectless state. But then, as you keep meditating on it, you realize that in fact your meditation still does involve a subtle kind of "sign" that is interpreted into a concept! The idea that serves as a sign in this case is a notion of time passing with no objects appearing in/to the mind! So while this meditation is closer to objectlessness, it still has a kind of very subtle object - the notion of time passing. 

So you progress to meditation on Nothingness. You just sit there with no content whatsoever. Your mind is completely blank. There is no perception of space or time. It looks like, finally, you have succeeded. However, once you master that, you realize that this absence of content in fact disappears when you stop meditating, therefore your meditation still has a limit, still has a focal point. You still artificially restrict your attention to exclude whatever is going on, and that counts as form. It's a very-very subtle object, it's an object of no-object - but an object nevertheless. The thoughts are stopped while you're in meditation, but once you come out of this withdrawal, you're back to square one. So this state is non-sustainable.

So then, as the final step, you realize that complete withdrawal from all objects is not as effective as learning to stop the associative mechanism from cycling. So instead of trying to generate a state with no objects, you transcend that restriction and allow your attention to stay wide open to stimuli, without apprehending any single one of them. Things may happen as normal but you don't associate, don't conceptualize, don't get carried away by them. Your meditation is truly formless (objectless), because the mind is wide open without grasping at signs. Unfortunately, this state is still useless because you can't really function in it. Any human activity that requires association and interpretation is incompatible with this method of liberation.

This ends the progression and concludes the study of semiotic perception.

The reason Buddha de-emphasized this approach, which apparently was known before Buddha, is because it has no direct relevance in his own framework, that of the Four Noble Truths. In Buddha's system the principle problem of sentient existence is "wrongness" (dukkha) and the goal is "suchness". So in Buddha's sequence of jhanas, the progression is focused on working with sources of "wrongness" in one's mental continuum, his method is to abandon coarse and then progressively subtler sources of "wrongness" while generating coarse and then progressively subtler states of "suchness" (joy=>bliss=>peace=>etc).

The formless jhanas are a nice exercise in concentration and an easy example of progressively refined meditative sequence that many practitioners in Buddha's times must have been familiar with, but in and of themselves they do not lead to Liberation as Buddha defined it - permanent, unconditional cessation of dukkha.

This is why most traditional sources describe formless jhanas as optional. In the ancient metaphorical language, the subjective experience of meditator in a formless jhana is compared with that of a deity at a certain level of heavens, whose life has no suffering but is finite and inevitably ends with eventual return to Earth.

While Buddha's jhanas are a progressive unfolding of his Four Right Efforts:


  "There is the case where a monk generates desire, endeavors, activates persistence, upholds &amp; exerts his intent for:
  "[i] the sake of the non-arising [anuppādāya] of evil, unskillful qualities that have not yet arisen.
  "[ii] ... the sake of the abandonment [pahānāya] of evil, unskillful qualities that have arisen.
  "[iii] ... the sake of the arising [uppādāya] of skillful qualities that have not yet arisen.
  "[iv] ... the maintenance [ṭhitiyā], non-confusion, increase, plenitude, development, &amp; culmination of skillful qualities that have arisen."  


And are a progressively closer approximation of the cessation of suffering declared in the Third Noble Truth.


  Now this, bhikkhus, is the noble truth of the cessation of suffering: it is the remainderless fading away and cessation of that same craving, the giving up and relinquishing of it, freedom from it, non-reliance on it.


In Buddha's four jhanas (known by contrast with the others as the "rupa" or form-jhanas), you examine your mind, identify the immediate sources of dukkha (cravings, attachments, hangups, mental conflicts etc.), as well as the various factors hindering your progress (distractions, pessimism etc.) and let go of those, while at the same time finding ways to generate the positive qualities that help you keep going, such as Motivation, Energy, Attentiveness, etc. as well as the approximations of suchness such as Pride, Joy, and Peace.

The culmination of rupa jhanas is liberation in the suchness of spontaneous presence, which involves no feeling of "wrongness" whatsoever and is endlessly sustainable because it is not limited by any condition other than the absence of aversions and cravings for things to be otherwise.

So unlike arupa jhanas which solve the wrong problem in the wrong way, Buddha's jhanas succeed by focusing on the right issue (that of the "wrongness" or "rightness" of one's experience) and then progressively refining the practice until it culminates in a fully sustainable, truly unconditional liberation.