Although the Anazazi left the Four Corners area, they most likely did not leave New Mexico. The current day Pueblo tribes are thought to be descended from the Anazazi. For an overview of the Pueblo tribes visit the New Mexico’s Tourism Department’s website: Pueblos, Tribes and Nations.
The Navajo and Apache are not related to the Anazazi. The Navajo and Apache moved into New Mexico around 1400 AD, migrating from the north. Instead, they are related to the Athabascan-speaking people in Alaska and Canada. The Navajo and Apache are still in New Mexico with many living on the reservations. The Navajo reservations are in the northern and northwestern parts of the state, and the Apache reservations are located in both the northern (Jicarilla Apache Nation) and southern (Mescalero Apache Nation) portions of the state.
The Spanish entered modern day New Mexico in 1581 by way of an expedition led by Franciscan monks Fray Agustin Rodrigues, Francisco Lopez, and Juan de Santa Maria. The three were accompanied by a military escort and a small number of servants. None survived to return to northern Mexico.
In 1595, another expedition into New Mexico from “old” Mexico began with the purpose of converting the native people to Christianity. This expedition was led by Juan de Oñate, 
who was expected to finance the expedition privately. In return for this private financing, the Spanish King named Oñate both Governor and Captain General, thus conferring on him the civil and military authority in New Mexico. The expedition actually started in January 1598, arriving at the present day city of Cuidad Juarez before advancing into what is now New Mexico. The expedition moved north, following the Rio Grande River. It stopped at each native village along the way to bring the people under the banner of Christianity and the Spanish King.
Oñate established the first capital of New Mexico at the confluence of the Rio Grande and the Rio Chama at the native village of Okhe. He named his capital San Juan de Los Caballeros; shortly thereafter, he moved to the west bank of the Rio Grande to the native village of Yunque and renamed it San Gabriel. San Gabriel served as the Spanish capital until Santa Fe was established in 1610.
In 1706, the present day city of Albuquerque was founded by Gov. Francisco Cuervo y Valdes. It was named after the Duke of Alburquerque who resided in New Spain (Mexico). The first "r" was dropped in the early 19th century, as noted in the 1807 narratives of the Zebulon Pike expedition.
The late 17th and early 18th centuries were a period of turbulence for the natives. A series of native revolts against the Spanish rule and Spanish reprisals took place that displaced many natives.
New Mexico, along with Mexico, won its independence from Spain in 1822. The early to mid 19th century saw the arrival of Anglos from the United States in the form of trappers and traders. This move resulted in New Mexico coming under the jurisdiction of the United States in 1848 under the Treaty of Guadalupe Hidalgo following the Mexican-American War. New Mexico remained a Territory of the United States until 1911 when it voted for, and was awarded, statehood.
Official State history website: http://www.newmexicohistory.org/home_html.php
The inhabitants of the great settlements in the Four Corners area of north western New Mexico, of which Chaco Canyon is the best preserved example, were descended from the Clovis people. They are now known as the Anazazi (the Navajo word for “the ancient ones”) and they inhabited the Four Corners region from about 200 BC to 1300 AD when their culture suddenly collapsed. The Anazazi first cultivated maize (corn) around the birth of Christ, and by the 13th century, they were cultivating maize, squash, and beans. They were also excellent pottery makers; their pottery had a distinctive black on white feature. By 1600, the Anazazi left their communities leaving us with a debate on the reasons why. Theories include drought, warfare, disease and even witchcraft.