This is a deceptively complicated question and requires a very difficult answer, which I probably cannot achieve here. I will try. I welcome all criticism and corrections.

"Pelasgian" is an ancient Greek conceptualization of the people who lived in Greece before the Hellenes (ie, the people who came to speak Hellenic dialects). This is fraught, because not all Greeks thought of their ancestors as transplants. The Athenians, for instance, asserted that they were autochthonous -- that is, that they had always dwelt in Attica since the beginning of time. It is not exactly a rigorous term, "Pelasgian," but modern scholars use it as a catch-all term for group(s) in the region from the Early Bronze age and earlier. I think it is very much out of fashion in 2019 in a linguistic or archaeological sense. One side note: "Pelasgian" is related tangentially to the Etruscans and the Etruscan language (of north central Italia), and I sometimes see it in lists with Etruscan language (as in the source I quote below). For your purposes I think it is best to just forget about the term "Pelasgian" altogether.

Historically, there have been several competing theories on who lived in "Greece" or the Aegean before the Mycenaeans (who spoke a language we now know was related to Greek and therefore Indo-European, and was written in a script called Linear B). We are roughly talking about the 2nd millennium BCE (but see below for a caveat on dating). Some theories call these people indigenes, while others do not. In my opinion, one theory in 2019 is by far the dominant, especially in linguistic evidence, and is probably the correct one: the "pre-Greek" people who lived in "Greece" were part of what we now recognize as the "Aegean substratum" stretching from Asia Minor/Anatolia to the Aegean, including Greece. It was once (19th century) thought that the Aegean substratum was non-Indo-European (aka "indigenous," before the arrival of the IE peoples into Europe). We know now that the Aegean substratum is in fact IE, and more specifically an Anatolian language, and more specifically Luwian. More and more evidence points to a koine, or region of common culture, from western Anatolia (Cilicia) to Crete to the Aegean to mainland Greece in the 2nd millennium BCE.

Ok. So Luwian is an Anatolian language, and we now know that the Anatolian family of languages is a branch of Indo-European. There are several members of the family: Hittite/Luwian/Palaic, Lycian, Lydian, and Carian, among others. Hittite wasn't even discovered until the Germans, in 1906, excavated in NE Turkey at a site called Boghazkoy and uncovered ten thousand clay tablets. They had discovered the archive at Hattusa, a once-powerful city of the Hittites, until then know only from Egyptian and biblical mentions.

All of these Anatolian languages, including Hittite, were in the past thought to be non-IE languages, but are now generally accepted to be IE. We can probably add Linear A, the enigmatic script of Crete, to this list (though the jury is still out). What this means for "pre-Greek" Greece is complicated, because a century of work was carried out under the assumption that there were non-IE peoples there before the coming of the Mycenaeans and then Hellenes:

Thus, with the progress of Anatolian studies, when it was shown that the languages spoken in both prehistoric and historic Asia Minor belonged to the Anatolian group, the presence of non-Indo-European speakers in the Eastern Mediterranean, formerly postulated for the entire region, became sharply reduced. As far as the present evidence goes, there are no visible traces of non-Indo-European speakers to the west of the Semitic languages of Syria and to the north of the Egyptian of Africa. Moreover, as Onofrio Carruba has shown in a recent article, Luwian is the only substratum language that can be traced west of a line drawn from the Bosporus in the north to the Gulf of Alexandretta in the south, that is, over the entire territory of Asia Minor. In view of these facts, it is hard to avoid the conclusion that the orthodoxy of the non-Indo-European pre-Hellenic substratum has lost its raison d’etre. (Finkelberg 2005, ch 3)

There is a lot of (mostly French) scholarship which has tried to find matriarchal vs patriarchal differences, ritual and cult variations, and so forth, all under this assumption. In essence, a lot of thought about these divisions has to be un-thought, and that process is still ongoing:

The significance of the new assessment of the linguistic identity of the Aegean substratum goes far beyond the field of linguistics. The case of matrifocality is the most conspicuous example. All the theories of matriarchy, beginning with Bachofen, proceeded from the evidence sup- plied by the historic peoples who had been associated with the ‘Aegean substratum’ – the Lycians, the Lydians, the Carians and so on. Yet, as we have seen, the linguistic evidence testifies to the Indo-European proven- ance of these peoples. Accordingly, the widespread practice of ascribing all the cases of matrifocality and related phenomena discovered among speakers of the conventional Indo-European languages to the influence of the non-Indo-European Aegean substratum should be revised together with the revision of the linguistic identity of the Aegean substratum itself. It is reasonable to suppose that such a revision will eventually lead to a more complex picture of early Indo-European society, a picture able to embrace both patriarchal and matriarchal Indo-Europeans, both nomadic and sedentary, both warlike and peaceful, and so on[.] (Finkelberg 2005, ch 3).

A note on dating: I've been saying, vaguely, "2nd millennium," but there is not a very good consensus for when exactly these IE/Luwian/non-indigenous people arrived in the Aegean. Some suggest their arrival at the cultural break between Early Helladic II and Early Helladic III (ca. 2200–2150 BCE). Coleman posited that 'proto-Anatolians' arrived in the Aegean about 4400 BCE. In Archaeology and Language (1987), Renfrew thought that perhaps the arrival of these people should be put back into the 7th millennium BCE, at the time of the Neolithic revolution. This is part of a broader theory of the spread of the first farmers to Europe from Western Asia (now understood to be IE peoples), and with them the spread of IE languages to the Aegean, the Mediterranean basin at large, and to northern and western Europe. I don't know very much about this topic, but the theory fits nicely with the linguistic evidence as it stands now.

That only gets us halfway to answering your question. There were people in the Aegean/"Greece" before the arrival of these Anatolian-language-speakers in the 2nd (or 5th, or 7th) millennium BCE. They are one more step removed from us compared to what we once thought, which only makes things more complicated. Can we get at them? Can we know anything about them? There has been at least one excellent recent attempt, which I've quoted from above. This is Margalit Finkelberg, Greeks and Pre-Greeks: Aegean Prehistory and Greek Heroic Tradition (Cambridge 2005). Here is the summary blurb:

"By systematically confronting Greek tradition of the Heroic Age with the evidence of both linguistics and archaeology, Margalit Finkelberg proposes a multidisciplinary assessment of the ethnic, linguistic and cultural situation in Greece in the second millennium BC. The main thesis of this book is that the Greeks started their history as a multi-ethnic population group consisting of both Greek-speaking newcomers and the indigenous population of the land and that the body of 'Hellenes' as known to us from the historic period was a deliberate self-creation. The book addresses such issues as the structure of heroic genealogy, the linguistic and cultural identity of the indigenous population of Greece, the patterns of marriage between heterogeneous groups as they emerge in literary and historical sources, the dialect map of Bronze Age Greece, the factors responsible for the collapse of the Mycenaean civilisation and finally, the construction of the myth of the Trojan War."

It's a super duper cool book, very smartly written, and absolutely cutting edge on this topic in this very ignorant layman's opinion.