A big part of Aztec religion was the concept of dualism. There were dual natures to many things, and the gods themselves had their own dual natures. A feathered serpent can fly as well as crawl on the ground.

In other words, Quetzalcoatl was fundamentally both the human figure and the feathered serpent at once. This kind of dual divinity was very important in Aztec religion. In some ways Quetzalcoatl was an aspect of the supreme creator, Ometeotl. The Toltecs and others associated Quetzalcotal with wisdom, as the personification of truth as a product of wisdom.

One story of Quetzalcoatl is about the creation of humans in the modern age (modern to the Aztecs). Man had been created the first ages of the world, but it was Quetzalcoatl who went to Mictlan, the land of the dead, to gather the bones of man and woman and bring them back to the Earth. Mictlantecuhtli, the aspect of the supreme being inhabiting the land of the dead, fought against him. Quetzalcoatl was killed, but rose again from death. He took the bones from Mictlan, and they were ground and placed in a vessel. Quetzalcoatl bled his penis on the bones, and he and the other gods did penance. It was through this penance that humans were given life (because this penance showed that humans deserved life).

The short of this is that the story represents internal struggle as much as it represents a recognition of the dual nature of everything. In other words, Aztec religion is incredibly complicated.

The book that I've taken all of this from is Aztec Thought and Culture, by Miguel Leon-Portilla.

So /u/barleystraw gave many of the basics of the worship of Quetzalcoatl among the Aztecs. I'd like to expand by explaining a bit of the deity's history.

Quetzalcoatl is almost as old as civilization in Mesoamerica. He was worshiped in Central Mexico as far back as the Formative Period, over 2,000 years ago. One of the main temples at the Central Mexican city of Teotihuacan was dedicated to him (and another deity as well). The aptly named "Feathered Serpent Compound" was a large priestly compound, and was likely the seat of the theocratic(ish) government of the city. He was a deity among many Mesoamerican cultures during the Classic Period, but his worship became more widspread during the Epiclassic period (800-1,000 AD) concurrent with the rise of Chichen Itzá in the Yucatan and Tula in Central Mexico.

He was worshiped by the Maya as Kukulkan. (Both Kukulkan and Quetzalcoatl can be translated into English as "Feathered Serpent".) The Mixtecs of Oaxaca worshiped him as "Lord Nine-Wind," and he was also an important deity to the Totonac culture on the Gulf Coast. In all cultures, he's a deity associated with life, nobility, and sacrifice, and is usually portrayed as a champion of humanity. /u/barleystraw gave the example of how in the Aztec creation story, Quetzalcoatl is credited with resurrecting humanity through an act of self-sacrifice following the destruction of the previous world. Among the Mixtecs, there was a semi-historical/semi-mythological conflict known as the "War of Heaven." This was likely an actual war, but it is described in the Mixtec codices as a war between mankind and supernatural entities from heaven and the underworld. During the conflict, Quetzalcoatl (as Lord Nine-Wind) intervened to fight on behalf of humanity.

As /u/barleystraw explained, he can be depicted as a humanoid figure or as the actual feathered serpent. Deities had the ability to adopt many forms. In addition to this, however, Quetzalcoatl was often equated with the wind deity Ehecatl. The two might be seen as two aspects of the same deity (in the same way that the Father, the Son, and the Holy Ghost are all aspects of the Christian God). In academic literature, he is often referred to as Quetzalcoatl-Ehecatl to distinguish him from Ce Acatl Topiltzin Quetzalcoatl, the semi-historical king of Tula who is also frequently called "Quetzalcoatl." Among the Aztecs, temples and shrines to Quetzalcoatl-Ehecatl were typically circular, likely to prevent the obstruction of the flow of wind around the temple.