How do I make "find" exclude the folder it searches in?

The easiest way would be to just add -mindepth 1, which will skip the first depth hierarchy and thus leave out your parent directory.

Also, you don't need an extra -exec call to rm, you can just delete the folders directly if they're empty.

find /var/www/html/content/processing -mindepth 1 -type d -mtime +1 -delete

If they're not empty:

find /var/www/html/content/processing -mindepth 1 -type d -mtime +1 -exec rm -rf {} \;

If you're lazy you can also have a wildcard expanded. Since * doesn't include the current directory by default (unless dotglob is set), you could also do:

find /var/www/html/content/processing/* -type d -mtime +1 -delete

However, this would also not include hidden folders, again due to the dotglob option.

The problem is that find returns the current directory (.) along with the other directories, so it deletes the processing folder as well as the subdirectories. A quick way to get around that would be to append the option

-not -name .

which stops find from outputting the current directory, and in turn stops it from being deleted.

That would work if you were running the command within the processing directory, so to allow for the fact that you are using an absolute path:

-not -name /var/www/html/content/processing

And the whole command would be:

find /var/www/html/content/processing -type d -mtime +1 -not -name /var/www/html/content/processing -exec rm -rf {} \;