A Seldon Crisis is a major turning point in the future of the Foundation; should the leaders of the Foundation make the wrong decision, the Foundation may fall, or be corrupted to the point of not being able to carry out the mission of creating a new Galactic Empire.
After each Crisis has concluded, a Vault opens and a hologram of Hari Seldon explains the significance of the Crisis the Foundation has just endured.
There have been several known Crises.
The first Crisis took place fifty years after Terminus was colonized; The sectors surrounding Terminus rebelled against the Empire, potentially cutting the Encyclopedia Galactica project on Terminus (aka the First Foundation) off from the Empire. Mayor Salvor Hardin succesfully solved the first Crisis by non-violently overthrowing the planetary government of Terminus and giving the technologically-regressing governments of the surrounding areas the excellent technology that only the Foundation could use. This created a balance of power; should any one of the four surrounding nations decide to take over the Foundation, the other three would ally against them.
The second Crisis came many years later, also under the watch of Mayor Salvor Hardin. This crisis was resolved through the proper application of the religious forces that the Foundation introduced into the surrounding nations as they provided them with technology.
The Third (probable) crisis came when the secular nation of Korell is given ships by a renegade leader from the Empire. Korell used the ships to launch attacks on the Foundation. The merchant prince Hober Mallow is instructed by the leaders of the Foundation to go look around and keep his eyes and ears open. Mallow proceeds to sell Korell a great deal of technology, all completely unencumbered by religion. Eventually, the leader of Korell goes to war with the Foundation, but is defeated by economic embargo, as the Korellians have grown too reliant on Foundation technology.
The Fourth (probable) crisis came when Bel Riose attacked the Foundation. He was on the edge of success, but then Emperor Cleon II called him and Brodrig to Trantor and had them executed on false charges, because they were becoming too powerful, and thus dangerous to his power.
The Fifth crisis was to be a civil war between the Foundation and the Independent Association of Traders, but this never occured due to the appearance of The Mule.
The Sixth and Seventh crisis are not mentioned.
The Eighth Crisis takes place at the beginning of Foundation's Edge, regarding the possibility of the Foundation moving its capital world from Terminus to a more central place, with Seldon advocating for continuation. The fact that Seldon's message fit the Crisis with such precision prompted Councilor Golan Trevize to theorize the Second Foundation was still alive and acting.