It's great to be able to communicate with so many people all over the world using the Internet. But it also means that there are more opportunities for defamation, or people publishing false and injurious comments that result in damages after lawsuits.
You can sue a website for the defamatory statements it posts to a limited extent. But Facebook can't be blamed for the inane things people say on it all the time. If someone uses the social media platform, or any other, to defame you, you can use that as evidence for your defamation suit against the insulting writer. And if, say, you are defamed on a personal or business website, then you can target the site in your suit.
The law protects people from both written and spoken statements that are false, harmful, and cause damage to a person's reputation. Written defamation is called libel. Spoken defamation is called slander. Both clams have similar elements, including publication or release to third parties.
In order to succeed in a defamation claim based on statements published online, you must prove that the statements were false and that they were published, thus damaging your reputation.
For public figures there is a higher standard applied to defamatory statements. People who are in the public eye can expect to be criticized and discussed extensively. And sometimes critics get it wrong. For a famous person to prove defamation they must show actual malice.
If you are being insulted online and the statements are false, make sure to collect the evidence. Posts and tweets can be deleted from social media, so take a deep breath, take a screen shot, and talk to a lawyer.
Do not reply to the comment ... unless you can muster a very simple "This is not true." And definitely don't insult the writer back. Do not attack the website or going around writing defamatory statements about the publication. Although it is not easy to do, you must keep your cool or risk finding yourself in the position of being sued, rather than suing, for defamation.