The term Anasazi is a Navajo word which roughly translates as "enemy ancestor" or the ancestors of our enemies. My understanding of the history of this region begins with the archaic indians who were nomadic hunter-gatherers (over 2000 years ago). The archaic indians gradually transformed into the basket maker culture which utilized pit houses in the 300-700 AD time frame. The Anasazi culture of cliff houses, kivas, and great cities (e.g. Chaco Canyon) flourished from about 700 AD until 1300 AD, at which time the culture seems to have abruptly disappeared. Current theories suggest that the Anasazi culture gradually migrated from their 4 corners strongholds to the Rio Grande valley and the Hopi mesas where they became today's pueblo and Hopi indians. The Navajo are believed to be recent immigrants from the north (based on linguistic similarities to the Athabascan Indians in Canada), having arrived around 1400 AD (after the Anasazi disappearance). An interesting fictional account (based on archeological evidence coupled with modern pueblo traditions) of life in Frijoles Canyon (now Bandelier National Monument) after Navajo arrival in the region is given in Adolf Bandelier's book, "The Delight Makers".
For the most part, dates of structures are determined by tree ring dating, "dendochronology". This is possible because tree rings form unique patterns indicating yearly conditions (drought, abundant rain, fire). Thus if we cut a 2 different 100 year old trees today in the same region, their tree rings will have approximately the same patterns. Now if we find an old dead tree which was near the end of its life when our recent trees were saplings, we can align the tree ring pattern for the years when both were alive and thus extend our pattern of tree rings back to when the older tree was a sapling. This has been done to extend the tree ring patterns back at least a 1000 years. Since a tree ring is added every year, this provides a very exact measurement tool. The Anasazi structures were built of rock with wooden beams. By examining the tree ring patterns in the beams, we can identify the unique pattern of the tree and tell exactly what year it was cut. Furthermore, the tree ring patterns give one clue as to the climate during the time the tree was alive. Once we know the age of the structures, archeologists assign approximate dates to the artifacts found inside. By identifying certain styles with a given age, it is possible to estimate the age of pottery found elsewhere, based on its style.
No one really knows. Toward the end of the Anasazi period they built and moved into the famous cliff houses which seem to provide great defensive capabilities, and yet there is little or no evidence of violent conflict. Abruptly around 1300 AD, following several years of severe drought, the Anasazi seem to have abandoned their cliff house dwellings and dispersed. The general consensus seems to be that their agrarian way of life had led to a population explosion, which coupled with poor farming methods had depleted the soil and other resources, just as a drought led to reduced harvests. As a result, the Anasazi left their cliff homes and moved to new territories, probably along the Rio Grande and on the Hopi mesas. Under this interpretation, today's pueblo Indians are the descendents of the Anasazi.
While linguistic analysis of modern pueblo languages indicates very little connection to central America, and hence suggests that both modern pueblo and ancient Anasazi migrated to the region from the north, Hopi legends tell a different story and fascinating tale. If I recall the details correctly, the Hopi believe their ancestors landed in Central America, arriving by boat from the west. After traveling north for some time, the different clans split up and agreed to meet sometime in the future. Each clan was to travel north, east, south, and west, before returning to a common meeting point. As the clans traveled (travel meaning to move somewhere and live for a number of years before moving on), they left their clan signs to indicate where they had been for others who might follow. Thus many of the petroglyphs are explained as indicating the passage of the clans (Kokopelli the flute player for the flute clan, snakes for the snake clan, and so on). During this lengthy travel they were responsible for building the Anasazi ruins we see today. Eventually the Hopi settled on what are now the Hopi mesas in Arizona, with the different clans arriving over a period of many years. This story which blends documented recent history together with legend into a continuous oral history of the Hopi people, ties the Hopi closely to the Central American cultures.
An interesting sidelight is the "for the most part" extremely non-violent nature of the Hopi. There have been several instances where different clans had serious and irreconcilable differences. Rather than resorting to physical conflict, one group would simply abandon their village and move away, or the two clans would engage in a "push of war" to resolve the conflict (perhaps with the loser leaving). This nature could explain the abandonment of the Anasazi dwellings without evidence of conflict. On the other hand, there is at least one instance in which the leader of a Hopi village decided that his people had strayed too far from their values and asked neighboring villages to massacre his village in a surprise night attack. For those interested in more information about the Hopi oral tradition, I highly recommend the "Book of the Hopi" by Frank Waters (Penguin Books 1977).