There are at least two ways to answer this question.
First, we can say that the Aztecs were the group of people who controlled an empire that was centered on the Valley of Mexico at the time that the Spanish conquistadors came to what is now Mexico. The Aztecs, at that time, had a large empire that they had gotten through conquest over the previous two centuries or so. They were the people, then, whom Cortez and the Spanish conquered to take Mexico as their own territory.
Second, we can say that the Aztecs were one particular ethnic group of Mexican Indians. They were originally from the northern areas of Mexico. They migrated down into the Valley of Mexico where they found a place that, they thought, their gods had told them to settle. They settled there and, over time, were able to take more and more power. They were able to gradually establish an empire by defeating the groups that had previously been in power in the valley.