I have a website and i would like to set up a minecraft server from there?

I have a domain from 1and1 and have seen that some people have got servers using website addresses rather than ip addresses. Is is possible to host a minecraft server through my domain?
If so, how?

If someone has a website for a minecraft server, it is likely just for info, forums, and/or to get whitelisted to get permissions to do things on the server. While it is possible to run minecraft from any port specified in server.properties file and included in the IP_or_name_of_server: port (instead of default port 25565), they are unlikely to run it from web port 80. You would need access to the virtual server (the operating system command line) or a dedicated server, with enough RAM to actually run the minecraft server (not just a web host with web space).

You'd need a Dedicated or Virtual Private Server (VPS). You could use the $20/month Linode plan to host a server for you and a couple friends (4-5-ish?) but if you get too many people on there, it'll start lagging, and you'll need a more expensive VPS. Linode has some of the best pricing on VPS around. Here are some instructions on setting up Minecraft Server on a VPS:

https://library.linode.com/web-applications/game-servers/minecraft-ubuntu12.04

After you get the server setup, you can point the domain name at it.