Things like IP addresses, domain details, hosting companies and payment processor accounts all play important roles but of them all, real-life names and physical addresses are considered investigation gold.
With these details in hand, a targeted and strongly-worded cease-and-desist notice may be all it takes to remove a service from the internet. Of course, these pieces of information aren’t usually left lying around.
With some luck, they may be available from companies providing services to pirate sites. Domain companies, Cloudflare and similar entities can be forced to supply this information in response to a DMCA subpoena filed by copyright holders. As a result, the studios of the Motion Picture Association and anti-piracy partners ACE regularly go to court in the US to obtain intelligence for use in investigations and legal action.