Let's pick an example for this article, so you can follow along... say TVN Solutions.
tvnsolutions.com has already been taken. So, go to a web solutions website, like tvnhosting.com
Type in the domain name you'd like with the correct extension (.com, .biz, .info, .net, .org, etc.) and click "go".
It'll say that "tvnsolutions.com" is already taken and next to that is a "click here for more info" link.
Alternatively, if you're not set on a name yet, it'll offer alternative names for your consideration below. But, as I have my heart set on "tvnsolutions.com" let's proceed by clicking on the more info link.
This will open up a second window that does a "Who Is" search on the name you're looking for. After a Notice and Terms of Use paragraph, you'll be presented with the registered name and address of the registrant of the domain name. You'll also get contact information of the registrant and below that you can see when their lease on the domain expiries.