The Aztec alphabet, in the sense of each symbol representing only a sound, was an introduction by the Spanish. Classical Nahuatl is the name of the language that was spoken by the people of the Aztec empire. The original Aztec language was not really written in an alphabet but a series of glyphs, as we will see in a moment. Nahuatl is also known as Nahua, Nahuat, and Nahual.
Read more about the Aztec language here!
The new Aztec alphabet
The Spanish introduced an Aztec alphabet based on Latin, basically the Roman alphabet many western languages use today. Since then, there's been a great deal of debate over how many Nahuatl words should be spelled and written. Still, the language is still alive in many parts of Mexico. It's estimated that there are over 1.5 million Nahuatl speakers in Mexico today! There are regional dialects, so there are many different ways to say simple words. As the renewed Nahuatl is being taught, a new way of writing it is coming into use, in an attempt to unite the various dialects with a common writing system.
The Aztec alphabet has 4 basic vowels: i, e, a, o
Consonants are usually written include: p, t, l, r, n, k, m, ch, x, c, s, z, ç, cu (qu), y, hu, p (these vary)
Confused? Read more about the Nahuatl or Aztec alphabet here, or the Various versions of the language. That way you can narrow down the alphabet based on a certain type of Nahuatl.
The Aztec alphabet as it was
The Aztec alphabet before the Spanish was not actually an alphabet but a series of pictures. How were these used? There were actually three different ways...
This means basically that the symbol meant exactly what it looked like. For example, this picture of a snake would simply mean "snake". Simple, yes. But to have a meaningful language hundreds and thousands of pictures would be needed (and there were a lot!).
These would represent the idea behind the symbol. So maybe a snake would represent a ruler (such as Itzcoatl - Obsidian Snake), or maybe footprints would represent a trip or the passage of time in a certain direction.
A phonogram is the closest we get to an Aztec alphabet. In this case, the picture actually represents a sound. Sometimes we do this just for fun in English - for example, you could use the picture of a bee to represent the word "be" (as in "to be"). If you say the word the picture represents, and put it together with others, you can get words and sentences totally unrelated to the objects the pictures look like.
In addition to the importance of the picture itself, colours were also important. When symbols looked very similar, they could easily be told apart by their colours. Of course, with modern technology and the availability of colour in computer screens and TV sets, colour is becoming more important in all languages around the world. Even on this page colour is used to split up various parts of the text to make it easier to read.
When telling a story, there wouldn't usually be a long line of glyphs, but rather a page with a few, positioned in such a way as to tell a story. It was kind of like looking at a scene in a story, a photograph of one moment. The glyphs would be used to remind you of various aspects of the story, and you would have to fill in the blanks.
Here's an example. Below is the symbol for "flint". If it just means flint, that's an pictogram. But it was also a calendar symbol, representing a specific day. Next is a combined symbol, tree and teeth. It means, literally, place with lots of avacados. The teeth are read as tlan, and ahuacatl is the avacado tree. The glyph actually means Ahuacatlan, a place name.
The Aztecs counted by 20s (as we usually count by 10s). They would use dots, as you can see on the Aztec calendar. Sometimes fingers would be used as well. 20 was a flag, then 20x20 (20 flags=400) was a feather or fir tree. 20x20x20 (20 trees=8000) was a bag or pouch (containing 8000 cocoa beans, commonly used for currency). Then, to show multiples of something, a line would be drawn to connect the numbers to the object. For example, below you can see a snake with 4 flags. 4x20=80, so the picture represents 80 snakes.
For more, read Aztec Books, Documents, and Writing, and this excellent article on the Aztec alphabet and glyphs.