To buy a domain, you need something called a registrar. A domain registrar is a reseller that 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 hosting or email services with the same registrar, so you can buy a domain and wait to find web hosting 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 hosting, which they are well acquainted with.
At most registrars, you can find out if a domain name is vacant or not, and also what top-level ones are available. Domainshop are very clear when it comes to both finding, buying and managing your domain.
When you buy a .no domain (ex: annec.no) you are required to be a Norwegian citizen or have a Norwegian organization number. You receive a self-declaration that you must sign before the domain is registered to you. The process is usually very fast and you will be notified that you are the owner usually the same day you order it.