Final answer:
The era described is the Age of Exploration or the Early Spanish Colonial Era, linked to Álvar Núñez Cabeza de Vaca's journeys in the 1530s, his work 'La Relación', and the consequent Spanish exploration and colonization activities in Texas and the broader New World.
Step-by-step explanation:
The era in Texas History being described, where Álvar Núñez Cabeza de Vaca documented Indian cultures and wrote 'La Relación', which in turn inspired the search for the mythical seven cities of Cíbola, is known as the Age of Exploration or the Early Spanish Colonial Era. Cabeza de Vaca's travels occurred in the early 16th century, specifically the 1530s when he and his crew were shipwrecked on the Gulf Coast and began their extensive interaction and documentation of the native peoples in what is now the American Southwest.
This period was marked by the Spanish conquest and colonization, where explorers sought new lands, wealth, and the spread of Christianity. The significance of Cabeza de Vaca's accounts contributed to the proliferation of Spanish expeditions and colonization efforts in the New World, influencing the future events such as the Texas Revolution and the creation of the Republic of Texas, known as the Lone Star Republic.
These historical transformations were part of a larger narrative of the Spanish arrival in the Americas and their confrontations and exchanges with the indigenous empires like the Aztecs and the Incas, leading to the establishment of New Spain and a period of cultural and political changes that have shaped the history of the region.