Aquire a specific domain name

I have a basic knowledge of the domain name system, however I have never bought a domain. I wish to acquire a very specific domain name for the sake of fleshing out my understanding.

Instead of acquiring a domain such as: 'websitename.com'

I would like something like: 'web.site.name'

I have browsed domain name providers and there seems to be some that break the '.com' mould such as '.horse' and '.cool'. But is there a way of getting a very specific domain name of your choosing or do you have to compromise?