It was likely an enemy of the Twenty Kingdoms (which are now the World Government), as implied by Professor Clover, and was seemingly destroyed utterly by them over 800 years ago at the end of the Void Century. In fact, it seems its defeat allowed the Ancient Alliance to seize global control and establish the World Government.
Furthermore, the ancient kingdom is the real reason the World Government has banned any Poneglyph deciphering and research, as the mere "existence and idea of this kingdom presents a threat to the World Government".
A brief illustration of the ancient kingdom was shown during Professor Clover's speech in the anime. It appeared as an island with huge marble buildings and vibrant green plantation. Apart from that, some moon-like craters are shown. The island was shown in a way that it was not clear whether it was in the sea or floating in the sky.
The "Great Kingdom" was first mentioned by Professor Clover amid the impending destruction of Ohara as he explained to the Five Elders that he knew why they are afraid of the events during the Void Century, as it would surely speak of this kingdom said to hold mighty power and ideals which, if discovered, posed a threat to the World Government.
About 800 years ago, the Kozuki Family invented a special kind of indestructible cubical monoliths, the Poneglyphs. The kingdom, likely aware of its imminent defeat and aware that its enemies planned to erase its existence from history, carved on the poneglyphs messages of special significance and scattered them in special locations throughout the world, thus entrusting them to future generations. The World Government banned any research of the Poneglyphs and methodically erased the information on the "Great Kingdom" from all historical records and texts. The Poneglyphs are now the only remaining source of information on the "Great Kingdom" or the Void Century.
The Poneglyphs so far have revealed information on the three Ancient Weapons, all of which are capable of mass destruction on a global scale. It may be possible that the people of this kingdom were somehow involved in the design and/or creation of these enormously powerful weapons. Again, this suggests that, as a people, they may have been significantly advanced in technology. However, the intended purpose of these weapons, as well as why the kingdom had built them has yet to be discovered.
It is unknown why knowledge of this kingdom is a threat to the World Government. Professor Clover seemed certain that the history of the kingdom could answer to that and other pertinent questions. The name of the kingdom itself was said to be of tremendous importance and it was said by Clover that it was the key of understanding the Void Century. However, before Clover could say the kingdom's name, he was shot. Further information has yet to be revealed.