Edit: Please, please read the answer before this whole thing, as it'll save you tons of time. Answering my issue was as simple as changing a setting with Outlook, it might be for you, as well. It had nothing to do with any of the config or log files, just the error.
So here's what's going on.
I can sign in with Outlook to my Mail Server, but when I go to send an email, I get this error. Using SquirrelMail, I don't get any errors, sending works fine. However for obvious reasons, I'm not going to use SquirrelMail.
Some files of potential interest are:
Mar 22 22:33:23 mail postfix/smtpd: connect from ool-45793236.dyn.optonline.net[126.96.36.199] Mar 22 22:33:23 mail postfix/smtpd: NOQUEUE: reject: RCPT from ool-45793236.dyn.optonline.net[188.8.131.52]: 554 5.7.1 <email@example.com>: Relay access denied; from=<firstname.lastname@example.org> to=<email@example.com> proto=ESMTP helo=<Ebear> Mar 22 22:33:26 mail postfix/smtpd: disconnect from ool-45793236.dyn.optonline.net[184.108.40.206]
nameserver 2001:4860:4860::8844 nameserver 2001:4860:4860::8888 nameserver 220.127.116.11 nameserver 18.104.22.168
So yeah, I have no idea what's wrong. By the way, the server I'm trying to send mail from is mail.ebear.us, MX records are configured correctly. Running Ubuntu 14.10
Thanks in advance for the help!
(This is a continuation of another question, a new one was requested by a different user as the main one was answered, but there's a new problem)