So here's my question: Is this happening because I didn't configure the DNS for that domain name on my server? If I configure BIND correctly for my domain name, will I then be able to browse through my website without a problem using the domain name?
Probably not because as first thing you need to route on the actual server.
I think you are being redirected to the router's IP or some other IP that is visible from outside (WAN , Internet) if this is not handled you will not browse the /var/www/some_dir_where_there_are_the_web-pages
but
Your_IP/someplace
on the computer that You use (the client) handle the hosts.
On nix this is usually /etc/hosts
On Win this is in WINxxx/System32/drivers/etc/hosts
put something like:
actual_LAN's_server_ip
www.my_domain.com
I have a public IP: 62.111.111.111
My Web server: 192.168.0.200
My client (browser): 192.168.0.100
thus the first 2 lines in my hosts are:
127.0.0.1 localhost
192.168.0.200
www.my_domain.com
Try. You should be able in seeing it.