What is my server name ?

You have to be more specific otherwise no-one is going to know which service (and by implication a server providing that service) you're trying to access. It's a bit like asking how long is a piece of string?

To help you and us get you the right answer (hopefully) what exactly is prompting you for the server's name?

Is it email? Or ftp? Or a calendar? Or perhaps files and/or folders etc?

Thanks Tom for posting but your instruction is for telling you the name of your computer. Although it can and sometimes is the server's name, more often than not it is not.

For example:

Most people using a Mac will have their computer - be it laptop or desktop - automatically named after the initial account is set-up, ie: Fred's MacBook etc. A server on the other hand won't and shouldn't follow the same procedure. When setting up a server an administrator (server or network) will probably be involved and will name the server something like mailserver, or fileserver or ftpserver etc. The domain name (eg: mydomain.com) will invariably be added to the name and the combination of name and domain becomes the hostname or FQDN (fully qualified domain name) of that server. I think this is what OP means by his/her question.

However you may be right? In which case until OP responds we won't really know. Will we?

Exactly Tom.

I'd like to take your instruction one stage further if that's OK? Assuming OP has access to the server in question and it is a version of OS X Server, the command should be run there. OP may think it has to be issued on their computer instead, but all this will do is tell him/her the name of their computer and not the name (hostname/servername) of the server.

Open System Preferences > Sharing > Computer Name

You're right. We do not know what he/she is trying to lookup.

To check for hostname. Open Applications > Utilities > Terminal, paste and enter

$ hostname

changeip -checkhostname