I need help with this anybody. I was experimenting with my Debian box trying to pass some control to my user account by chgrp /bin/* to the "staff" group. After this the "su" would always say

su: Authentication failure
Sorry.

And ping would tell me:

wongdg@dg-debian:~$ ping 202.188.0.133
ping: icmp open socket: Operation not permitted

Changing them back to root group didn't help either. There must be ways to set this right. I couldn't find anything relevant on this subject in this site. So, would really appreciate anyone able to contribute.

Many thanks,
Wilson

Check the permissions on the file /bin/su. It should be owned by root/root and have the SUID bit enabled in order for it to work for users.
There aren't very many things which need the SUID bit enabled, but /bin/su is one of them that does.
check 'man su' and 'man chgrp' amd 'man chmod' for more info.
Let us know if that solves the problem.
For ping, the same applies, but first you could see if it is executable by everybody (--x--x--x) or even executable at all (as root type chmod +x /bin/ping). I think Ping by default also has the SUID bit enabled.
Beware of anything with the SUID bit enabled, it can lead to security holes.