The Leon Klinghoffer/Achille Lauro episode* in the middle 80s set off a wave of statutes pushed through to allow US prosecution of people for attacks on Americans abroad, based solely on the status of teh victim as a US person, or on the ship or airplane being US registered.

*  You remember - a couple putative terrists took over a cruise ship and launched Holocaust survivor Klinghoffer off the fantail in his wheelchair, while people watched. When they were ultimately taken into custody and brought before an Italian court (IIRC, the liner was registered in Italy), that court found no jurisdiction over crimes on the high seas and released the putative terrist.

Another aspect of extraterritorial jurisdiction is in statutes which provide for prosecuting US citizens for crimes committed abroad.  The most recent prominent example is the torture trial in Miami of a man (whose name escapes me) relative to his involvement in the Liberian civil war, where he was a member of a prominent political family.  The defendant got 147 years for torture;  his link to the US was his dual citizenship here and there.

Another example is the Foreign Corrupt Practices Act, which criminalizes US persons and companies giving bribes overseas (so as to get business or government favors in that foreign country).

Then there is the different concept of extraterritorially kidnapping someone to bring them back to the United States for trial.  This was approved in 1992 by a 6-3 Supreme Court (Rehnquist opinion, Stevens, O'Connor and Kennedy in the dissent) in the case of United States v. Alvarez Machain, holding that an extradition treaty with another country, in that case Mexico, does not bar the United States from kidnapping a citizen of that country and putting him on trial in Federal court.

FROM THE DECISION:
The issue in this case is whether a criminal defendant, abducted to the United States from a nation with which it has an extradition treaty, thereby acquires a defense to the jurisdiction of this country's courts. We hold that he does not, and that he may be tried in Federal district court for violations of the criminal law of the United States. Respondent, Humberto Alvarez Machain, is a citizen and resident of Mexico. He was indicted for participating in the kidnap and murder of United States Drug Enforcement Administration special agent Enrique Camarena Salazar and a Mexican pilot working with Camarena, Alfredo Zavala Avelar. The D.E.A. believes that respondent, a medical doctor, participated in the murder by prolonging agent Camarena's life so that others could further torture and interrogate him. On April 2, 1990, respondent was forcibly kidnapped from his medical office in Guadalajara, Mexico, to be flown by private plane to El Paso, Tex., where he was arrested by D.E.A. officials. The District Court concluded that D.E.A. agents were responsible for respondent's abduction, although they were not personally involved in it.

Interestingly, but unsurprisingly, in one of Yoo's memos about torture not being torture and other contortions, he cited Alvarez Machain for a much more expansive view of how the government can act - that the Fourth Amendment does not apply overseas, among other things.  It's an example of how precedents can be stretched by inventive sophists.  Yoo provided those in spades.

Then there is rendition - of the ordinary type.  

This, as my prior comment noted, is the process of sending the person from one jurisdiction to another to stand trial in the other jurisdiction's courts.  The person would have been charged in Jurisdiction A and captured in Jurisdiction B, either pursuant to a regularly issued warrant out of Jurisdiction A which had been domesticated in Jurisdiction B or following an arrest in Jurisdiction B for breaking B's local law.  Domesticating a warrant would mean undertaking to make sure that a warrant out of "A" would be valid for "B's" police to arrest and hold someone in "B".  Most extradition treaties have some sort of provision to make this process pretty much automatic.

An example.  Assume a defendant wanted for murder in a capital-punishment US state, who fled the US to Europe and then, while in (say) England, was arrested by the local police for dealing in controlled substances.  One of two things would likely happen.

Either he would pop up on a warrant list issued by that US state and put into the flow of warrants from everywhere, and the UK police would see that.  They would then notify the US Department of State of holding a US person and the existence of the warrant showing up.

Or the UK police would notify the US consul of having arrested a US national (assuming they, unlike Texas cops dealing with Mexicans they arrest, obey the Vienna Convention) and the US consul would see his name on a list of warrants.  The word of the arrest would work its way back to the local US prosecutor (who'd had the warrant issued).  The local prosecutor, to get this person extradited from the UK (which forbids extraditing people to places where they are exposed to capital punishment) so he could be tried for the murder, would have to issue a written, binding promise (often backed by a court order) that this defendant would not be exposed to capital punishment if sent back.  This would pass through the federal government to the UK government.  Once the UK was satisfied, they would be free (once the defendant had completed his UK sentence to their satisfaction - often they'll allow a prisoner to finish his sentence at home) to extradite him to the US.  

The actual shipping of this person would be the rendition.  

That is ordinary rendition, in the international context.

In the inside the US domestic context, rendition is the process of physically shipping a defendant pursuant to an extradition order (or if he waives extradition) back to the state which issued the warrant for his arrest.

Extraordinary rendition is what Obama has banned.  FWIW, he could not ban ordinary renditions, as those are set up and operated pursuant to treaties (extradition treaties) with foreign nations.  While (arguably) a President can pull the US out of a treaty without consulting or consent by Congress, it has to be done explicitly  (if only so the other country knows).  That was not done here.  So, extradition treaties, which provide for shipping (rendering) people from one regularly-constituted court to another in a different country, continue to operate.  Even for the Bush family and their Paraguay estates (we now have an extradition treaty with Paraguay).

I hope this (very simple) explanation clears up any confusion.