If you have an email address that ends with something like xtra.co.nz or yahoo.co.nz, your email address is actually provided by someone else, such as your Internet Service Provider (ISP).
If you want your own shorter email address, you could register your own .nz domain name and use that to set up an email address. For example, if you registered anyname.nz, you could have john@anyname.nz.