Salafism is a branch of Sunni Islam, who believe not only in Islamist government, but Islamic rule based on the Qur'an and the first generation of interpretation. They reject not only secular government, but modernism; they want to reestablish the Caliphate. The movement is heterogeneous, however, Salafis diverge with respect to the final form of the Caliphate and how that end state is to be achieved.
In most current discussion, Salafism refers to 19th and 20th century formulations of the ideal, although it can trace back to the twelfth century ideas of Ibn Tamiyya. Many of the violent armed jihadist movements of today are Salafist, especially al-Qaeda, but not all Salafists are violent and not all violent Muslim extremists are Salafist.
Salafism draws from the Arabic word salaf, or "to precede", which refers to the al Salaf al salih, the virtuous fathers of the faith who were the companions of the Prophet. These fathers learned Islam directly from the Prophet; Salafis to bring Islam back to that perfection, using a method, manhaj, that verifies if actions conform to the Qur'an and the Sunna as confirmed by the Hadiths. The last salaf was Ahmad Ibn Hanbal, who founded the Hanbali school of Islamic jurisprudence. 
There are Salafists that reject recourse to violence despite their opposition to foreign influence and the state. Saudi Wahhabism does not demand a return to the time of the Prophet, so Saudi Arabia disapproves officially of Salafism.
Modern Salafism is not monolithic; factions differ on the nature of the end state (i.e., the caliphate), and how to reach it. While it includes the conservative Wahhabi clerics of Saudi Arabia, it also includes more militant Sunni of Islamists.
Most Sunni jihadist thinkers draw legitimacy for their actions from the Salafist ideology, but they have differences in whether the "near enemy" (i.e., Islamist states that do not properly enforce Islamic law) or the "far enemy" of non-Muslims are the first priority for overthrow. Influencing their thinking are clerics including:
There are arguments that al-Qaeda subscribes to Wahhabi Salafist doctrines, but not all Wahhabis subscribe to al-Qaeda.
It is intensely opposed to all forms of shirk (polytheism), and subscribes to the doctrine of Al-wala’ wa’l-bara’, love of things Muslim and hate of things that are not. Muslim, here, is used in a strict Salafist definition.
- Brian M. Drinkwine (January 26, 2009), "The Serpent in Our Garden: Al-Qa'ida and the Long War", Carlisle Papers, Strategic Studies Institute, U.S. Army War College
- Basheer M Nafi (9-15 January 2003), "The root of all evil", al-Ahram Weekly
- Juan Jose Escobar Stemmann (September 2006), "Middle East Salafism's Influence and the Radicalization of Muslim Communities in Europe", Middle East Review of International Affairs (MERIA) Journal
- Quintan Wiktorowicz, The Management of Islamic Activism: Salafis, the Muslim Brotherhood and State Power in Jordan (New York: SUNY, 2001), p. 112, quoted in Stemman