Companions: Susan, Barbara, and Ian
Written by: John Lucarotti
Directed by: John Crockett
Background & Significance: After the success of long-lost Doctor Who adventure "Marco Polo", script editor David Whitaker asked that story's writer John Lucarotti to write another historical epic about whatever he chose. Lucarotti chose "The Aztecs", a culture which had fascinated him, and they were off to the races.
While it is considered one of the great stories of the classic series, and very memorable in terms of Hartnell's tenure on the show, to call "The Aztecs" a wonderful epic story of awesome is... hyperbole. It's best defined by two very important events that happen over the course of the adventure:

The second bit we'll laugh at when we get to it in the story, but "The Aztecs" does (to an extent) deal with the threat of changing the course of history. In this, "The Aztecs" is, perhaps, most important. The idea that Barbara can, through the aid of The Doctor bringing them to that moment in history, alter the past to fit her worldview and design is nothing short of revolutionary in the scope of the series. Now, The Doctor wouldn't be just having to run away from people who want to kill him and saving his companions (or having them save him). This story established that he needs to have history play out as it should play out and that they can't go and change anything.
(Anything, of course, being rather subjective as a term.)
Really, though, that just makes this story sound much more epic than it actually is. It's not like Barbara goes around and shows the Aztecs guns or anything. She just tries to change their culture in a very small way, which, while not massive and epic on the scale of what other characters (from Time Lords to humans to Daleks) will try to do later on in the series, is at least the first story to tackle the issue, thereby giving them all the license to do it in the future.
So let's get to it!