Zaidism is the oldest surviving branch of Shia Islam, and the one that has kept the least of what the word Shia now suggests to most people.
It takes its name from Zayd ibn Ali, a grandson of Husayn, who revolted against Umayyad rule at Kufa in 740 and was killed. What the revolt established was a doctrine of the imamate quite unlike the one that developed among the Twelvers. The imam is not designated in an unbroken line and does not simply inherit the office. He is any qualified descendant of Hasan or Husayn who is learned, pious, and rises openly to claim the office and fight for it. Quietism disqualifies: a man with the lineage and the learning who does not rise is not the imam.
The consequences run through the whole system. There is no infallible imam and no doctrine of concealment: no awaited figure, no occultation, no clerical deputyship exercised in his absence. Zaidi theology was shaped by the rationalist Mutazila, and its jurisprudence is close enough to Sunni practice, and to the Hanafi school in particular, that Zaidism is routinely described as the Shia school nearest to Sunnism. Zaidis do not curse the first caliphs, and Zaidis and Sunnis in Yemen have prayed in the same mosques for centuries.
It is sometimes called the Fiver school, on the reasoning that it stops at the fifth imam where the Twelvers continue to the twelfth. The label is an outsider's convenience rather than the tradition's own, and it misdescribes a school whose distinctive claim concerns how an imam is recognised rather than how many there have been.