To buy a domain, you need something called a registrar. A domain registrar is a reseller who has an agreement with NORID to provide domain services. There are many registrars and many also offer web hosting and email services.
You do not need to have domain and web space or email services with the same registrar, so you can buy a domain and wait to find web space until you are in contact with the people who will create your website. This is often an advantage, as web developers/designers often have their preferred web space, which they are well acquainted with.
With most registrars you can find out if a domain name is free or not, and also which top level is available. Domain shop is very clear when it comes to both finding, buying and managing your domain.
When you buy a .no domain (e.g. annec.no), you are required to be a Norwegian citizen or have a Norwegian vat registration number. You will receive a self-declaration that you must sign before the domain is registered to you. The process usually goes very quickly and you are told that you are the owner usually the same day as you order it.