First at all, a domain is the unique “name” that you can register on internet for easy find a webpage or webapp. Maybe on this times sounds a quiet obvious but not always was like this. So with this clarified, we can proceed to the next question: How check if a domain is available?
Well its easy, with a domain provider. You only need to put the domain to check (ex: www.mygreatdomain.com) and the provider tells if the domain is available and how much cost it.
Domain providers
None of these providers pay me to promote them, I simply used them in the past and the experience was good: