I am going to assume you are referring to the Inca and Aztec civilisations of South America? In that case, I will plough ahead and give you a reasonable response for the fall of both empires.

Firstly, it is important to note that these cities were very advanced and were praised as such, particularly in Hernan Cortés' letters to Charles V. He praised Tenochtitlan in these letters, so it is clear that the civilisations of South America were quite sophisticated, although when measured against our own moral standards they fall quite short, but that comes with the study of history...

Firstly, the Cortés' expedition led to the discovery of Tenochtitlan which subsequently fell in 1521. Cortés had exploited the fact that Tenochtitlan's military expansionism had led to it developing a number of enemies. Along with these enemies, such as those of Tlaxcala, Cortés captured the Aztec leader Montezuma and through him, he captured the city. Tenochtitlan consequently fell in 1521. Francisco Pizarro too arrived during a period of unrest within this time the Inca empire, and in 1530 his expeditionary force apprehended the Inca ruler Atahualpa, who had killed. Without their leader, the Inca empire withered away and its capital, Cuzco, fell in 1530. Therefore the fall of these empires is easily attributed to civil unrest within these two civilisations.

Next, the population deteriorated rapidly due to ravaging disease. Since everybody has a general idea why, I'll throw in some statistics just to show you how drastically the population decreased. The indigenous population of central Mexico fell from 25 million in 1521 to 2.6 million in 1568. In Peru, it fell from 3.3 million in 1520 to 1.3 million in 1570. The diseases included smallpox and measles which destroyed the populations due to their lack of immunity.

The policy of encomienda was also a factor in the transposition of native and coloniser. It began as a form of lordship where a 'deserving person' could receive payment from indigenous towns and villages. However, it was soon more related to a form of slavery. They ranged in size, Cortés had up to 115,000 indigenous workers across 23 encomiendas. This oppressive dominance over the indigenous population was another factor in the dwindling of population. In less than a century their whole way of life was completely transformed.

Along with these, there were other factors such as the use of horses (which the indigenous population had interpreted as god-like) and sheer ruthlessness (massacres, etc).

I have to study, so I'll wrap this up with Bartolomé de las Casas' own words on the treatment of the indigenous populations by the Spanish Conquistadors:

'The common ways mainly employed by the Spaniards who call themselves Christian and who have gone there to extirpate those pitiful nationsand wipe them off the earth is by unjustly waging cruel and bloody wars. Then, when they have slain all those who fought for their lives or to escape the tortures they would have to endure, that is to say, when they have slain all the native rulers and young men (since the Spaniards usually spare only the women and children, who are subjected to the hardest and bitterest servitude ever suffered by man or beast), they enslave any survivors. With these infernal methods of tyranny they debase and weaken countless numbers of those pitiful Indian nations.' - Bartolomé de Las Casas, Brevissima relacion de la destruycion de las Indias (Sevilla, 1552).