But before the file can be edited, it's ownership will need to be changed from System to your user. Control-click on the mime.types file, and from the contextual menu choose Get Info. Then, in the Details section of Ownership & Permissions, click the padlock icon to unlock the section. An administrator password will probably have to be entered. Then, just change the Owner from system to the user-administrator. Leave the window open; you'll use it again shortly.
Now, the mime.types file is ready to be edited. You can use TextEdit, or any other pure text editor, to edit the file. Near the bottom of the file, change video/mp4 mp4 m4a m4p to audio/mp4 mp4 m4a m4p. Then, using cut and paste (Apple-C and Apple-V), move that line so that it is between audio/mpa-robust and audio/mp4a-latm. Save the changes. And, as above, change the ownership of the file back to system!
For the changes to take effect, the server will have to be re-started. To do that, just go to System Preferences and choose Sharing. On the Services tab, click-to-highlight Personal Web Sharing, and then click Stop and Start. That's it. AAC files will now play as music in Apache Server.
[robg adds: In my mime.types file, I didn't have a multi-item video item. My line just read video/mp4 mp4, so I just commented it out, and added the above line in the audio section. I also used the Terminal, saving the ownership steps, via sudo vi /etc/httpd/mime.types, but TextEdit is definitely easier to use than a Terminal editor. Note that this hint will not let you listen to a protected AAC file for which you don't already have access; it just tells the web server what to do with AAC files when it finds one -- pass it on to QuickTime, I believe.]

