root@Slack:/# find / -name *.new
/usr/share/texmf/tex/latex/titlesec/titletoc.new
/usr/share/texmf/doc/latex/fancyhdr/fancyheadings.new
/tmp/src2pkg-1.8/tar-1.13/scripts/weekly.new
/tmp/src2pkg-1.8/Resources/etc/DONT_PURGE.locales.new
/tmp/src2pkg-1.8/Resources/etc/src2pkg.conf.new
/tmp/src2pkg-1.8/src2pkg-1.8-i486-1_K26/etc/src2pkg/DONT_PURGE.locales.new
/tmp/src2pkg-1.8/src2pkg-1.8-i486-1_K26/etc/src2pkg/src2pkg.conf.new
/etc/ssh/sshd_config.new
/etc/ssh/ssh_config.new
/etc/kde/kdm/kdmrc.new
/etc/rc.d/rc.bind.new
/etc/rc.d/rc.font.new

What do I do with these .new files?
The ones in /etc are the ones I got after running upgradepkg

The ones in /tmp can probably be deleted, but you might check them like the others first. The others need to be compared to the file they are supposed to replace, ie compare /etc/ssh/sshd_config.new to /etc/ssh/sshd_config. If you have made any custom modifications to the original, modify the .new file appropriately, then copy the .new file to the original. I usually like to move /etc/ssh/sshd_config to /etc/ssh/sshd_config.orig before I move the new one over.
Regards,
Bill

On gentoo you've got etc-update (to mention one tool)
The generic way: diff -u oldfile newfile | less
probably take a look at 'man diff' for better or more advanced options.
</very basic comment> :P

slackroll and slackpkg provide ways of automatically detecting and dealing with these files, and yes, the usual thing is to run a diff on both files (hint: vimdiff), the old one without the .new suffix and the new one with the suffix, and either leave the old config file alone or port the old configuration settings to the new format, if needed.