The ideological origin, and the name, of the Wahabbi movement come from Muhammad ibn Abd al-Wahab an 18th century preacher whose predecessors were Ahmad ibn Hanbal (d. 855) and ibn Taymiyyah (d. 1328).

Basically having learned a strict, tradition (i.e. hadith) based form of Islamic legal reasoning he found it difficult to reconcile those beliefs with the popular religious practices as he found them in both his place of birth in the Najd as well as in Basra or any of the other places he traveled. He viewed things like saint worship as idolatry and Sufistic practices as "bida," heretical innovation. Upon returning to his hometown he gained the support of the local amir but was eventually expelled.

He found refuge with Al Saud. The prominence of his movement has been linked to the success of the Saudi family ever since. But it has not been a direct and soaring ascendancy. After his death his work was carried on by his descendants, the Al ash-Sheikh who retain extraordinary power within Saudi Arabia to this day. But there were mis-starts. The partnership of the Al Saud and Abd al-Wahhab resulted in the First Saudi State which expanded rapidly. However their practices and assaults on the pilgrimage caravan to Mecca raised the ire of the Ottoman empire, who, holding the caliphate, were guardians the pilgrimage. The first Saudi State was then destroyed following a long campaign in which they were repulsed from the Hedjaz and their ruler Abdullah ibn Saud, being executed in Constantinople.

Without getting into the 20 year rule, the modern prominence of the movement, other than, again, the link between the Saudi and ash-Sheikh families and their governance of a state with some 30 million people is that Saudi Arabia has for some time been committed to spreading their particular form Islam for decades by funding mosques, printing and distributing conversion literature, free Qurans etc. etc. such that it's estimated they are responsible for ~90% of such funding.

The result is that you can probably walk into just about any Sunni mosque in the world and find some piece of religious literature that was paid for or produced by Saudi Arabia.

While there has long been radical currents of thought within Islam, which have come to prominence at different times in history, much of modern Sunni Radicalism has it's origins in Egypt in the mid 20th Century. The Muslim brotherhood was the first modern political party (and some would say the first modern terrorist group) to advocate against secular government in the Middle East and "Western" values. Sayyid Qutb, a religious scholar, became a leader of the Brotherhood in the 1950's and 60's and formulated much of what we know think of as radical Islamic ideology. His ideas were a reaction to the pollution of Middle Eastern culture and its gradual political subjugation by France and Britain during the dissolution of the Ottoman Empire.

They were also a repudiation of secular nationalist governments in the wake of their failure to bring widespread economic prosperity to the region, their inability to defeat Israel, their continued collusion with Europe and the US and the failure of the Pan Arab movement to politically unify the Arab states. Qutb was an Islamic Utopian. He believed that the application of Isalmic law to every area of public and private life would bring a complete political, cultural and economic transformation to the Middle East. He was part of a conspiracy to assassinate President Nasser and overthrow the secular government of Egypt; he was imprisoned and eventually executed in 1966.

In the environment of war, civil strife and deep economic inequality that has characterized the Post-War Middle East, these ideas have resonated very deeply and given rise to both largely peaceful political movements and terrorist groups.