The name (like other TeX packages) is supposed to be a pun and is intended to be pronounced like "Aztec".
In addition, it handles the organization of files and references in a convenient way. My years of experience working on books and journals led to this (easily modified, modular) system...so I can't give you all of its features in a brief description. Instead, let me list a few things just to give you an idea:
For example, if you are writing a textbook, you will likely have exercses at the end of every chapter or section. ASTeX provides special commands for typesetting these exercises and automatically generates a solutions manual. No, it doesn't write the answers for you, but it takes the answers you have written (and included in each exercise file) and puts them together in the right order and with the right numbers. As a result, it is very easy to move, delete, add or reorder questions.
Similar nice features exist for the articles in a journal. Special commands which load them automatically generate a table of contents (with title and author name), generate running heads and put a "box" on the first page of the article with journal information. Each issue of the journal automatically starts on the page following the last page of the previous issue.
One thing which is required for a professional looking product is consistancy in the graphics. ASTeX contains many commands to ensure this consistency. In particular, the commands ensure that the line weights are the same from picture to picture, that the labels on figures are always the same distance from the object they refer to, that the arrows used to point to objects are always the same, that shaded regions are consistent, etc. Moreover, it does all this in a way which is very convenient to the user. In fact, you can scale a figure merely by changing one variable, and all of these things will remain consistent!
For more information, check out the documentation.
Journals and research texts typeset by ASTeX include The Pacific Journal of Mathematics, The Journal of Differential Geometry and Proceedings of the Second IFT Workshop: Yukawa couplings and the Origins of Mass.