With all due consideration,

Taking into account the original Pali terminology, mindfulness [sati] and concentration [samadhi] are best explained within the teaching of the perfectly enlightened one with regard to the seven enlightenment qualities and the noble eightfold path.

1. Firstly, mindfulness as a quality of true enlightenment [satisambojjhanga] and concentration as a quality of true enlightenment [samadhisambojjhanga] are to be made to arise and to be developed to fulfilment, just as instructed for all seven qualities. So in a sense a difference can be said to exist between the way in which each quality manifests itself, although their essence is always the same, namely true enlightenment.

2. Secondly, right mindfulness [sammasati, having the same doctrinal meaning as the concept of vipassana] and right concentration [sammasamadhi, sometimes refered to as samatha] are the 7th and the 8th steps of the noble eightfold path, or the path that progresses towards the eradication of suffering. Right mindfulness is actually the brief exposition of the development of mindfulness [satipatthana] while right concentration explains the four steps of attainment [jhana] for the eradication of suffering. In no case can they be defined or understood as different methods or ways of practice. They can be said to differ only as being different steps of the only correct method for attaining perfect freedom of mind.

In other words, mindfulness and concentration are not separate or different meditation techniques. On the contrary, they are both qualities of true enlightenment and developing sati is the progressive process, while attaining samadhi is the progressive result and confirmation of the one and true path that leads away from mental misery and anguish, towards mental purity and emancipation.

Using a simile for the question, one might ask if walking on a path towards a destination (right mindfulness, sammasati) and arriving at milestones along the way until the destination is reached (right concentration, sammasamadhi) can occur separately. The answer is definitely no, one can not occur without the other, especially when they are understood correctly.

Respectfully

@Suminda's answer is quite thorough, so I'll just give some few extra pointers.

"I am new to buddhism and meditation (I have been practising it for one month now), and despite finishing my fourth book on the matter, I am still very unclear about what a meditation session should look like when practicing either concentration or mindfulness."

One popular way to start your meditation practice is to follow the instructions found in the Ānāpānasati sutta. One thing one can do is dedicate meditation sessions to the first practices described, just to start. For example:

“Breathing in long, he understands: ‘I breathe in long’; or breathing out long, he understands: ‘I breathe out long.’ Breathing in short, he understands: ‘I breathe in short’; or breathing out short, he understands: ‘I breathe out short.’ He trains thus: ‘I shall breathe in experiencing the whole body of breath’; he trains thus: ‘I shall breathe out experiencing the whole body of breath.’ He trains thus: ‘I shall breathe in tranquillising the bodily formation’; he trains thus: ‘I shall breathe out tranquillising the bodily formation.’

Alternatively, one can dedicate meditation sessions to, say, the observation of body and feeling as described in the Satipaṭṭhāna sutta.

These practices, as it turns out, help with the weakening of the five hindrances (restlessness, sloth/torpor, sensual desire, aversion and doubt). These hindrances are critical obstacles to overcome. So weakening them should probably be the focus of the practice until one is able to be free from them for some reasonable period of time.

So, at this point, all the effort should go into understanding and suppressing the five hindrances. Part of this practice is done in the day to day, part by the practices above, and part by developing concentration (which is also helped by the mindfulness meditation).

Much of the practice of meditation at this stage is popularly taught as simply calming and "letting go", but they benefit significantly from the "mindfulness" practices.

"From what I understood, each of these practices are clearly differents and even have different pali names (Samatha and Vipassana if I am right ?)."

Some traditions and teachers draw a clear distinction between them. Some ignore "concentration" altogether (likely as a shortcut, but sacrificing one branch of the eightfold path; this is a polemical subject), and others teach concentration in a very deep and "absorbed" way and do "vipassana" once they emerge from absorption.

A lot of these interpretations come from the exegetical texts, in particular, from the classic book Visuddhimagga -- "Path of Purification".

The suttas, however, allow for other readings. For example, in many discourses, right mindfulness is said to precede right concentration:

[...] In a person of right mindfulness, right concentration [comes into being].

-- AN 10.103

Also, there are some interesting parallels between the jhanas (meditation states that correspond to the "right concentration" of the eightfold path) and the Satipaṭṭhāna ("mindfulness"). So, it's possible to read the suttas and understand these practices going hand in hand and deeper, together.

For example, we know that being free from the five hindrances are pre-requisite to the entering the first jhana. Moreover, being secluded from the hindrances creates conditions for some special experiences of happiness and rapture, which are jhana factors.

Now, if we look at the second part of Ānāpānasati, we see that the meditation objects are these experiences of rapture and happiness and it's about further tranquilizing certain activities (possibly vicara/vitaka, to attain second jhana):

“He trains thus: ‘I shall breathe in experiencing rapture’; he trains thus: ‘I shall breathe out experiencing rapture.’ He trains thus: ‘I shall breathe in experiencing pleasure’; he trains thus: ‘I shall breathe out experiencing pleasure.’ He trains thus: ‘I shall breathe in experiencing the mental formation’; he trains thus: ‘I shall breathe out experiencing the mental formation. ’ He trains thus: ‘I shall breathe in tranquillising the mental formation’; he trains thus: ‘I shall breathe out tranquillising the mental formation.’

So, likely this part of the practice is better performed once the meditator is free from hindrances and can get to a certain tender and gentle state where these factors are present.

Moreover, these factors themselves contribute to the unification ("concentration") of mind. They also "retroactively" make the five hindrances weaker in our lives.

I am wondering why concentration and mindfulness are referred to as disctinct practices in the books I read, as it doesn't make much sense to me. At this point I am afraid that I may be missing something important.

As I understand it, this stems from how certain traditions came to understand and do these practices.

Mindfulness is the cause for concentration. The suttas say:

Of those, right view is the forerunner. And how is right view the forerunner? In one of right view, right resolve comes into being. In one of right resolve, right speech comes into being. In one of right speech, right action... In one of right action, right livelihood... In one of right livelihood, right effort... In one of right effort, right mindfulness... In one of right mindfulness, right concentration comes into being... In one of right concentration, right knowledge... In one of right knowledge, right release comes into being.

MN 117

Mindfulness means to bring to mind & maintain in the mind the Teachings. The suttas say:

One is mindful to abandon wrong view & to enter & remain in right view: This is one's right mindfulness.

MN 117

Dwelling thus withdrawn, one recollects that Dhamma and thinks it over. Whenever, bhikkhus, a bhikkhu dwelling thus withdrawn recollects that Dhamma and thinks it over, on that occasion the enlightenment factor of mindfulness is aroused by the bhikkhu; on that occasion the bhikkhu develops the enlightenment factor of mindfulness; on that occasion the enlightenment factor of mindfulness comes to fulfilment by development in the bhikkhu.

SN 46.3

Concentration or samadhi means 'collectedness' or the 'gathering' of the mind's spiritual faculties, as follows:

The Blessed One said: "Now what, monks, is noble right concentration with its supports & requisite conditions? Any singleness of mind equipped with these seven factors — right view, right resolve, right speech, right action, right livelihood, right effort & right mindfulness — is called noble right concentration with its supports & requisite conditions.

MN 117

In summary, mindfulness called 'The Gatekeeper' (such as in SN 35.245) acts to keep the mind free from unwholesome states. As mindfulness maintains this clarity, due to increasing purity, the mind's clarity gathers energy into a stability & focus of this clarity, which samadhi/concentration.

The term 'vipassana' means 'direct/clear seeing' and is a direct result of concentration rather than a direct result of mindfulness. Mindfulness is the cause of concentration and concentration is the cause of vipassana. Thus the suttas say:

In one of right mindfulness, right concentration comes into being... In one of right concentration, right knowledge comes into being... In one of right knowledge, right release comes into being.

MN 117

Thus this Noble Eightfold Path comes to fulfilment in him by development. When he develops this Noble Eightfold Path... these two things — serenity (samatha) and insight (vipassana) — occur in him yoked evenly together.

MN 149