There's a lot of personal info we'll remove such as images of minors, involuntary fake pornography and other issues as outlined here: https://support.google.com/websearch/troubleshooter/3111061
Our announcement this week was about an expansion of removing personally identifying information -- PII -- (such as email addresses, physical addresses, phone numbers, etc),. This is the page that explains in more detail and where people should begin the removal process: https://support.google.com/websearch/answer/9673730
The removal process will lead you to a troubleshooter. That's designed to help people more quickly get things resolved. For example, if a web page no longer exists on the web, or the site owner removed info, there are faster options you can use rather than going down the formal removal request option.
But let's say you want something removed that has PII from a page that's live on the web and where you for whatever reason don't want to contact the site owner. The troubleshooter from that page I mentioned will lead you here: https://support.google.com/websearch/troubleshooter/9685456#...
Say No to contacting the site owner, select Personal Info from the next option, pick the type of info and you'll get to the form (such as here if it involves contact info): https://support.google.com/websearch/troubleshooter/9685456#...
That's where you enter the info. Appreciate the feedback that we should look at how to improve this more -- that said, the troubleshooter itself isn't new, and people have used it to successfully remove information beyond our recent expansion. But we'll see how to make it better.