How to publish my website onto a server?

Do you have a hosting service? If not, you'll need to sign up for one. Most of them charge between $3-10/month for basic plans. The hosting service will undoubtedly let you access your allocated space via FTP, for which you'll need an FTP client such as Cyberduck or Transmit.

 

Depends what kind of site it is (HTML pages/Flash?) but most hosting packages allow you to upload files via their control panel to. That's if downloading and setting up the FTP software is something you'd want to avoid. Not that is complex but if you're not going to upload stuff much i'd use the control panel route.
If you're with a semi-decent host they'll have loads of help how to do this. I'd recommend inmotionhosting.com if you haven't already got hosting as they have all this as standard.