The American Censorship link includes a lot of information about what this US law would change and how it would affect user-generated content sites like Stack Overflow.

Currently, if someone posts copyright material to Stack Overflow, there is a well-established legal procedure (called the DMCA) that establishes how the copyright owner can get that material removed. That law, while imperfect, has done a great job of balancing the interests of copyright holders and websites that host user-generated content (like us).

So for example, right now we receive takedown notices from professors demanding that we remove technical information because they claim that students are using it to cheat, people who claim that we are posting questions that they use in job interviews, and so forth. If we took down everything somebody wanted us to take down, the Internet would be worse. Right now, under the DMCA, we require the person making the complaint to send us a complete DMCA take down notice. This must include, among other things:

contact information

a statement that he has a good faith belief that use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

a statement that the information in the notification is accurate

a statement that, under penalty of perjury, the filer is authorized to act for the copyright holder

a signature

This requirement deters an awful lot of people who are randomly trying to censor Stack Overflow for hosting answers that they don't find convenient.

We respond by notifying the person who posted the material, giving them a chance to make a case for why the material is non-infringing.

The important thing about DMCA is that if we follow this fairly decent procedure, we, as a website, are legally protected from the claim that we contributed to the copyright violation.

The SOPA dramatically alters the careful balance in favor of "alleged" copyright holders. It makes it impossible for websites to find a reasonable safe harbor allowing them to continue to host user-generated content. And that is life-threatening for websites like Stack Exchange.

See also: Text of the bill - PROTECT IP / SOPA Breaks The Internet on Vimeo - SOPA infographic