In the United States, there isn't a strict analogue for your Censor Board. Here, the First Amendment to the US Constitution provides protection to film makers, ensuring the government won't censor their work. However, the law doesn't prevent voluntary censorship, which is where the MPAA comes in. They're a trade group of movie studios who created a ratings system. Films can voluntarily be submitted to be rated by them.
Movie theaters in the US then do some enforcement of policies based on these ratings. For example, usually children under 17 are not allowed into rated R films unless accompanied by an adult. In addition, theaters will sometimes not screen any films that lack a rating. Theaters will also often refuse to screen NC-17 films, which ends up acting as a sort of censorship, as when a film is being made in part to make a profit, not being able to distribute it to thousands of theaters nationwide due to an NC-17 rating means the contents of the film will be self-censored down to rated R-levels of content.
Usually in the making of a film, a specific rating will be targeted for marketing reasons. So a film that the studio wants to have broad appeal will have its content crafted to ensure a PG-13 rating, which doesn't have the "no people under 17 can freely get in the theater" limitation that rated R films have. This helps ensure more people can easily watch the film in theaters.