"Mycenaean Greece" is synonymous with Late Helladic Greece. "Helladic" is used to refer to ceramic (i.e. pottery) phases in mainland Greece during the Bronze Age, divided into Early, Middle, and Late Helladic periods. Because the Late Bronze Age in mainland Greece is characterized by what archaeologists have defined as "Mycenaean" culture (named after one of the main centres in Late Bronze Age Greece, the city of Mycenae), Late Helladic is often referred to as "Mycenaean" instead.
It is important to note that the designation "Mycenaean" is archaeological, not ethnic. While it is customary to refer to the inhabitants of the LBA (= Late Bronze Age) mainland as "Mycenaeans", this can be misleading: we don't know for sure how the people of the Bronze Age Aegean referred to themselves. As such, we cannot know if e.g. people from Thessaly thought they were "one people" with those from the Argolid, let alone with those from the islands or Crete. Crete is especially interesting, because of its earlier "palace" culture -- if the court complexes such as those unearthed at Knossos, Malia, and Phaistos are indeed "palaces", which seems doubtful! -- characteristic of the Middle Bronze Age (referred to as "Middle Minoan" on Crete). After ca. 1400 BC, there is evidence that Knossos, the main centre on Crete, had been taken over by people from the mainland and served as kind of a capital city for the whole of Crete.
One of the defining features of Mycenaean Greece, especially during the Late Helladic IIIA and IIIB periods (ca. 1400-1200 BC), are the palaces. A "palace" in this context is thought to have been the place where the kings of the Mycenaean cities resided. We know from the Mycenaean tablets -- written in a script referred to as "Linear B" -- that the king was referred to as a wanax, i.e. "lord". Each palace formed the centre of a small(ish) individual kingdom, usually tucked away in a high fortified place (i.e. a citadel), like at Mycenae, Tiryns, and other places.
The focal point of each palace was a rectangular room with central hearth and a porticoed porch in front of it. This basic structure is referred to as the megaron and is thought typical of Mycenaean culture. The central room contained a throne, as attested at Pylos. On Crete, we find a megaron at, for example, Agia Triada, during the period when Crete was under the influence of the mainland. (This is also when we encounter other structures on Crete that are typical of mainland i.e. "Mycenaean" culture, such as chamber tombs.) The megaron is essentially the groundplan for the Greek temple of the first millennium BC; the Mycenaean wanax may have played an important role in LBA religion. If you're interested in this development, check out Alexander Mazarakis Ainian's From Rulers' Dwellings: Architecture, Religion and Society in Early Iron Age Greece (1100-700 BC), published in 1997.
When someone refers to "palace culture" in the context of the Bronze Age, they refer most likely to the Mycenaean period. The focus is then also specifically on the palaces themselves; there were regions in mainland Greece where no palaces have been found. "Palace culture" in this sense refers to the material culture that is thought characteristic of the palaces, so megarons, chamber tombs, tholos or beehive tombs, wealthy grave goods (e.g. silver cups), objects made in particular styles, the use of wall-paintings, and so on.
Guy Middleton, in his book The Collapse of Palatial Society in LBA Greece and the Postpalatial period (2010), writes that
Mycenaean palaces society [...] may have been made up of a network of elites within and between regions. Palace states may have existed as ‘centres of alliances and systems of subordination and dependence’ (Dickinson pers.comm.). In the Argolid, with its high concentration of major sites, this may be especially true but links between palace states and non-palace state areas may also be expected [...] and some palatial interest in the Aegean islands, especially Rhodes, also seems likely.
The Eastern Mediterranean was thoroughly interconnected during the second millennium BC (I.e. ca. 2000-1000 BC) in particular, with interconnections growing ever more dense during the Late Bronze Age. The Aegean, Anatolia, the Levant, Cyprus, and Egypt were in constant contact with each other. Elites played an important part in these networks. Rulers exchanged gifts with each other, for example, as demonstrated by wall-paintings in "Minoan" Crete that depict monkeys and cats (from Egypt), and the presence of alabaster and ivory objects in the Aegean, and so on. "Palace culture" played an important role in these exchanges. Shipwrecks, such as that found at Uluburun, also give an idea of the kind of trade that went on in this period.
The term "Postpalatial" refers here to the period after ca. 1200 BC, when most of the Mycenaean centres have been destroyed (though many are soon reoccupied), and when trade and exchange in the Eastern Mediterranean suffered a severe setback. It should be noted that these destructions did not happen overnight; it was a process that took decades, so the idea of a single event bringing an end to the Mycenaean palaces, as often suggested in popular accounts, is misleading.
I hope this helps. In addition to what I've already cited, I recommend:
I also write regularly about the Bronze Age Aegean on Ancient World Magazine.
Please feel free to post follow-up questions.