A cytokines storm, or cytokine cascade as more normally called in clinical settings, is just an over agonism of cytokines. The cytokines activate a large number of lymphocytes, more specifically t-cells. This over activation is what causes the harm, as too many cells aggregating in one location can cause tissue damage depending on the location.
Free radical scavengers, such as dimethylsulfoxide, can freely bind to cytokines and reduce their effect but treatment is difficult. No medications are known to prevent it.
Anything that excites cytokines can activate a cytokine cascade. Ebola, the flu, etc.
If this is not enough detail I'm sure you could find a lot of information at the national library of medicine division of the national institute of health website. Good luck.