Self-radicalization is a phenomenon in which individuals become terrorists without affiliating with a radical group, although they may be influenced by its ideology and message. It usually addresses radical Islam, but there certainly have been instances of "lone wolf" terrorism from all ideologies, such as Timothy McVeigh and Eric Randolph on the U.S. right.
Self-radicalization does not mean the eventual terrorist activity never affiliates with a group, or even takes place in the country of radicalization. that Many attacks considered completely "home-grown" have had assistance, but the participants came together in the West. Mohammed Atta and three others directly involved in the 9-11 Attacks radicalized while in Hamburg, Germany, and then traveled to Pakistan. The 2005 London bombings were committed by British citizens, although subsequent investigation suggested that after radicalization, they may have obtained training from al-Qaeda, and then returned. 
Many had seen the U.S. as relatively immune to this, due to societal characteristics, and indeed it had been seen, in various forms, in Europe long before recent U.S. cases of Islamist radicalism. Peter Bergen observed,
In sharp contrast to Muslim populations in European countries like Britain -- where al Qaeda has found recruits for multiple serious terrorist plots -- the American Muslim community has largely rejected the ideological virus of militant Islam. The "American Dream" has generally worked well for Muslims in the United States, who are both better-educated and wealthier than the average American. More than a third of Muslim Americans have a graduate degree or better, compared with less than 10 percent of the population as a whole.
For European Muslims there is no analogous "British Dream," "French Dream," or, needless to say, "EU Dream." None of this is to say that the limited job opportunities and segregation that are the lot of many European Muslims are the causes of terrorism in Europe -- only that such conditions may create favorable circumstances in which al Qaeda can recruit and feed into Bin Laden's master narrative that the infidel West is at war with Muslims in some shape or form all around the world. And, in the absence of those conditions, militant Islam has never gained much of a U.S. foothold -- largely sparing the United States from the scourge of homegrown terrorism. This is fundamentally a testament to American pluralism, not any action of the American government. 
Bergen and others observed that the U.S. was not immune to its citizens, such as John Walker Lindh, going abroad to join terrorist movements. In these cases, the self-radicalization led to joining a movement, rather than conducting an act of terrorism at home.
A clearer case seems to have been associated with the 2009 Fort Hood shootings by Nidal Hasan.  He was born and educated in the United States, and, while in electronic contact with Anwar al-Aulaqi, did not leave the U.S. or, as far as is known, associated with any groups. He received only two messages from al-Aulaqui and was not under his operational direction. 
Timothy McVeigh, convicted and executed for the 1995 Oklahoma City bombing, was a U.S. citizen who became radicalized over the Ruby Ridge and Waco incidents, but was actually rejected by domestic right-wing groups he tried to join.
Misperceptions of self-radicalization
Most of the participants in the 2004 Madrid bombings were first thought to involve self-radicalization, but it now appears that most were radicalized before the 9-11 Attacks and the Iraq War, but the actual group coalesced in Spain. Most were economic immigrants although one was a native Spaniard. Internet access complemented face-to-face group formation, allowing downloading of such things as the works of Sayyid Qutb and Abdullah Azzam.  Indeed, "a gradual sense of progressive involvement, usually considered a consistent quality among those becoming terrorists, seems to be absent for some latecomers in the Madrid bombing network. 
- Bruce Hoffman, Challenges for the U.S. Special Operations Command posted by the Global Terrorist Threat: al-Qaeda on the Run or on the March?, Terrorism, Unconventional Threats and Capabilities Subcommittee, House Armed Services Committee
- Peter Bergen (19 November 2009), "The Terrorists Among Us: Why an al Qaeda attack on U.S. soil is still a real threat.", Foreign Policy
- Spencer S. Hsu (18 November 2009), "Hasan's ties to radical cleric raise issues for law enforcement: Imam's rhetoric on Web fell short of triggering legal action, officials say", Washington Post
- Juan Zarate, "The Fort Hood Attack: A Preliminary Assessment," testimony before the Senate Homeland Security and Government Affairs Committee (November 19, 2009), http://csis.org/files/ts091119_zarate.pdf
- Fernando Reinares (November 2009), "Jihadist Radicalization and the 2004 Madrid Bombing Network", CTC Sentinel, Combating Terrorism Center, United States Military Academy 2 (11), pp. 16-19
- John Horgan, “From Profiles to Pathways and Roots to Routes: Perspectives from Psychology on Radicalization into Terrorism,” Annals of the American Academy of Political and Social Sciences 618:1 (2008): pp. 80-93, cited by Reinares