Your question is more complicated than it may first appear to be. If you are attempting to run a mail server, (which it sounds like you are) then you'll need to tell us which distro and which mail server you are using. Thunderbird is an email client, it sends and receives mail to/from email servers. Relaying is when you try to send an email message through another server to the destination server; some people will use their ISP as their email relay server because their ISP's email server is typically not in a dynamic IP range and not flagged as SPAM immediately. However, you don't have to do this type of relay if your ISP doesn't block port 25 (and you still don't even if they do, but it's much more complicated if they do). If you are running your own server, you can send directly to the other email server.
Anyway, onto your apparent problem. Your MX record is how email servers identify other email servers. When you do a lookup on a domain (using something like nslookup or dig) you can see the MX record in their DNS entry. If you are trying to send email, most receiving servers expect you to have an MX record for the domain you are sending from. If you have a hosting provider, you probably already have an MX record. If you don't, you may want to think about looking at a different host, or learning more about how DNS works to run your own DNS server(s).
In either situation, you will need an MX entry in your DNS record to relay email to most hosts.