I have a PHP5 app on RedHat ES4 and Apache 2. The app accesses images in a /files/images/img/ directory. The /files/images/ structure is wide open with permissions at 777. The img directory and the files themselves are at 750. The images are owned by 'nobody' with the same for the group ('nobody'). I put apache in the 'nobody' group but the app still cannot read the images; I get a permission denied error in the httpd log. I am misinterpreting how permissions work with the Apache webserver and files?

hmmm... i am not an apache expert but i think you have to define
alias /images/ /files/images/img/

<Directory "/files/images/img">
AllowOverride None
Order allow,deny
Allow from all
</Directory>

section in your httpd.conf, if your image directory is not in the same dir where your DocumentRoot is.
and what are your /files permissions? if they are set to non readable for all except the owner and your apache user is different that the owner, then the apache user won't be able to enter that dir, if am not mistaken.
also check the permissions of your files (not only the directories).