Search for any domain and you get results for that domain but not every indexed page. For that, you use the site: operator. That is the purpose of the site: operator.
No, I get that the idea is pretty much true, but I think that it is important given the confusion surrounding the site: operator results to really be specific.
Doing a search for a domain name "example.com" without the site operator will do a search using the domain name as the keyword, it will search across the entire Google index and return pages from a variety of place. Say you a Facebook page for your website, then that will likely be returned. Say spammers are scrapping your content and linking to your website, that could be returned, say your competitor has a blog post explaining how much better their product is vs yours and your domain is in the text, then that may appear.
Doing a search with the site: operator limits to the search to results from the that domain. BUT (this is the important distinction) it does not return all the indexed pages for the domain. It returns a subset of them. This subset can include all pages, typically in the case of a small site, but there is no guarantee. And the search "site:example.com" says return everything, but you can also do a search like "widget site:example.com". This will return only pages that match widget. This is what happens when you sue the search box here WW. It is also worth pointing out that if you repeat site: search you may get a different subset.
And the final point (a point that often causes confusion) a site: search returns, at the top of the Google SERP, the number of pages found, this is NOT the number of pages from your website in the index, it is only the number of pages returned by the search. As a above it may be the number if your site has few pages, but it likely is not, instead it is likely a number less than the true number and even that isn't a certainty.
