Popping in to clear up something. (And I can't believe I'm making this as my first post after returning here)
(And by clearing up, I mean copy and pasting an explanation of the bill from a lawyer-in-training I know in my city)
To understand the background behind SOPA, you have to understand the rationale that arose to create the DMCA. This will be incredibly simplified, but don't worry about the details. Basically, back in 1996 the WIPO passed two copyright treaties, and in 1998 the US decided to pass the DMCA to implement those treaties (dealing with criminalizing infringement mechanisms) and at the same time deal with a major problem that was already beginning to arise: that of the publisher-infringer.
See, under the copyright act of 1976 (I think, I forget the year), publishers of infringing content are also liable. So, for example, Youtube would be liable for content published. Obviously, internet publishers (like ISPs, hosts, etc) didn't want this. Unlike traditional publishers they had no real ability to screen all of their customer's content, and didn't want to be obligated to do so. At the same time, getting content down was expensive and difficult; they had to sue, go to court, get an injunction- and with the rise of the internet, they might not be able to figure out WHO to sue, or even know that that person was located in a suable jurisdiction.
Hence the DMCA created a compromise. Publisher-hosts (ISPs, Youtube, even though it didn't exist yet, etc) were granted 'safe harbor' as long as they obeyed the takedown requests. The takedown requests allowed copyright holders to take down infringing content without suing or without finding out where the people who put the content up were. That saved them (and the people who put the content up) the expense and trouble of a lawsuit. And, the counter-notification process required that the person provide their full name an address so that they could theoretically be sued if it went that far.
Essentially, the DMCA created a compromise regime- it was faster, safer, cheaper, and basically more efficient for everyone. It didn't stop the copyight holders from suing- it just made it so they mostly didn't want to because this was more efficient. On the flipside, most infringers didn't get sued because their content was removed quickly. Now, to be fair, the primary objection to the DMCA regime was that by making it easier for corporations to push these takedown notices, they would start taking down material they didn't have the copyright to or for which some affirmative defense existed (like fair use), and that nobody would counter-takedown for risk of lawsuit, and so it would have a chilling effect.
Over the past 13 years, the primary issue for the copyright holders has been that the DMCA doesn't let them target hosts which are overseas. For example, say I run metube.com out of Belize. The copyright holders could send all the DMCA takedown requests they want, but my host (Belize Internet Services) doesn't give a crap about the DMCA. They won't respond to the takedown notice. And at the same time, metube.com is of course available from anywhere on the internet.
So the content holders went back to Congress, and said "Look, we need to be able to target metube.com." The way they chose to do that is SOPA. It contains a number of different provisions, but the two that are relevant here are S.102 and S.103 (they're the most important sections of the bill anyway). They allow for two different mechanisms of targeting foreign sites.
S.102 is the seizure rule. It allows the United States Attorney General to make a request to the US District Court for an order to seize the DNS of a website if:
(1) the Internet site or portion thereof is a U.S.-directed site and is used by users in the United States;
(2) the owner or operator of such Internet site is committing or facilitating the commission of criminal violations punishable under section 2318, 2319, 2319A, 2319B, or 2320, or chapter 90, of title 18, United States Code; and
(3) the Internet site would, by reason of acts described in paragraph (1), be subject to seizure in the United States in an action brought by the Attorney General if such site were a domestic Internet site.
It also allows the USAG, on receipt of this order, to 'take down' the same things as are authorized under S.103 (internet advertising and payment processing), and allows the USAG to tell search engines to delete the site from their registers.
That's so much legal chatter, but basically what it means is that the Attorney General can seize foreign sites which are committing or facilitating criminal IP infringement if those sites are targeted at US users, and would be seizeable if they're hosted in the US. The last point, (3), is actually relevant, because under 18 USC 981 and 18 USC 2323, the Attorney General can already seize websites if they're based in the United States. And it has done so in the past, and continues to do so. It's worth noting, though, that the USAG doesn't target Fanfiction.net. It doesn't target Spacebattles.com. Its primary target is actually counterfeitters- domains like nfljerseysupply.com, handbag9.com, and dvdprostore.com. A few other sites have been targeted- large torrent hosters, primarily. Given that the provision in S.102 basically mirrors the domestic seizure provision for sites outside the US, it seems unlikely that the Attorney General is going to suddenly go on a spree of seizing websites with its new powers.
S.103 is the secondary takedown power. People tend to confuse the two, feeling that anyone can get a DMCA-like takedown request to take down entire websites under SOPA, but that isn't really the case. Anyway, S.103 basically creates a DMCA mirror provision, but rather than targeting publisher-hosts, it targets payment network providers (b)(1) and internet advertising services (b)(2). Exactly the same thing, basically: a copyright holder can make a request to one of those two services, saying "This site is an open and notorious infringer of copyright". Under that section, a payment processor is required to freeze or deny transactions between the site and US customers, and an IAS is required to stop advertising to or from (i.e., stop putting advertisements on their website and taking money, and stop putting advertisements to their website on other websites) the infringing site. There is, also, a counter-takedown procedure, just like in the DMCA.
Basically, SOPA creates two primary tools: 1) It expands the seizure rule for domestic sites to foreign ones (and adds a few more powers that domestic seizures don't have, like blocking off IAS and payment processors) and 2) it creates a DMCA-like takedown procedure that IP holders can use against sites which are outside the US and therefore can't be targeted directly. The rationale here is petty simple: most payment processors (Paypal, Visa, mastercard) are based in the US, and most internet advertising services that people in the English-speaking world are likely to see are also based in the US (google, primarily).
*Breathe* Wow, that was long. I hope it helps, though?