In the old days relaying mail was no problem and most servers accepted mail for any destination. But with increasing spam activity the setup has been changed and requires authentication. That means that you have to use an existing mail account at the SMTP server (your client domain mail id), connect to the server using an authentication method, and send the password for your account.
You are probably already using authentication with your mail clients (Outlook, Thunderbird, etc). Just use the same parameters as there. Note that a different port may be used.
In general use something like
smtp.UseDefaultCredentials = false;
smtp.Credentials = new System.Net.NetworkCredential("yourusername", "yourpassword");