The Stop Online Piracy Act (Sopa) is the bill being considered by the House of Representatives.
The Protect Intellectual Property Act (Pipa) is the parallel bill being considered by the Senate.
The proposed legislation is designed to tackle online piracy, with particular emphasis on illegal copies of films and other forms of media hosted on foreign servers.
The bills propose that anyone found guilty of streaming copyrighted content without permission 10 or more times within six months should face up to five years in jail.
The US government and rights holders would have the right to seek court orders against any site accused of "enabling or facilitating" piracy. This could theoretically involve an entire website being shut down because it contains a link to a suspect site.
US-based internet service providers, payment processors and advertisers would be outlawed from doing business with alleged copyright infringers. Sopa also calls for search engines to remove infringing sites from their results - Pipa does not include this provision.
The bills would also outlaw sites from containing information about how to access blocked sites.
The bills originally demanded that internet service providers block users from being able to access suspect sites using a technique called Domain Name System (DNS) blocking.
This would effectively make them "disappear" from the internet - and is a process already used in China and Iran. However, after opponents claimed this could disrupt the internet's underlying architecture, the chief sponsor of each bill agreed to ditch the measure.
To protect sites against false claims of illegal activity Sopa proposes penalising copyright holders who knowingly misrepresent a site's activity - however, Pipa does not contain this safeguard.
Both bills offer immunity to ISPs that block access to websites if they have "credible evidence" that the third party's pages contain unsanctioned copyright material. Critics claim this could create a conflict of interest as it may encourage firms to block access to competitors' sites.
It could also encourage firms to take a "safety first" approach resulting in users being prevented from viewing legal material.
Sopa's supporters are trying to reach consensus on the bill before putting it to a vote in the House of Representatives, which suggests that any vote may be some way off.
Senate majority leader Harry Reid plans to put Pipa up for a vote in the upper house on 24 January.
Supporters of the bills include television networks, music publishers, movie industry bodies, book publishers and manufacturers.
Critics include Google, Facebook, Twitter, Wikipedia, Yahoo, eBay, LinkedIn, AOL and Zynga.