The suffix identifies the name as belonging to a specific top-level domain (TLD). As of late 2016, when this article was written, there are numerous TLDs available for general purchase, including .com, .edu, .game, .green, .hiphop, .net, and .org. The most popular of these by far is .com, which is supposed to indicate commercial sites, but in reality has come to include almost everything.
The dispute was over the innocently-named Site Finder Service, launched by Verisign in 2003. With traffic to every .com domain going through Verisign, the Site Finder Service redirected anyone accessing an unregistered domain (via a typo, for example) to a Verisign website with sponsored links. The company argued that this was more useful to Internet users than seeing a generic error page. Whatever their reasoning, this made Verisign the de-facto owner of every unregistered domain (what else does it mean to own a domain?), and skyrocketed them from 2,500th into the top 10 most visited websites. The best part? Verisign launched this without running it past anyone.
In the first quarter of 2015, 294 million domain names had been registered. A large fraction of them are in the com TLD, which as of December 21, 2014, had 115.6 million domain names, including 11.9 million online business and e-commerce sites, 4.3 million entertainment sites, 3.1 million finance related sites, and 1.8 million sports sites. As of July 2012 the com TLD had more registrations than all of the ccTLDs combined.
If you're having trouble finding a domain name (whether because of crowding or cyber-squatters), check for a help facility on each registrar's site. Domain registrars typically house search engines that return a listing of available names similar to the one you want. When you search for a domain name at Namecheap, for example, you get both the status of that name and a list of suffixes available for that name. Maybe [Sitename].com isn't available, but [Sitename].biz or .org is.