The part of a domain name that is after the first dot is known as the domain extension. For example in the domain name yourdomain.co.nz, the domain extensions, or TLD (Top Level Domain), is .co.nz
You can have as many versions of your domain name with various extensions as you want, provided that they aren't already registered. In fact, it is considered a good idea to register several traditional extensions to protect yourself against cybersquatters.
You should be aware however, that there are a few domain extensions that are restricted, and only available to only certain types of people. For example, you must be a NZ educational institution in order to register a .school.nz domain, and a .com.au domain will ask for an ABN (Australian Business Number) before you can complete registration.