"The Protoss are such powerful psychics that they can inadvertently send out psychic "ripples" that are disruptive to other life forms. With careful training, High Templar learn to focus these ripples into a Storm of raw psychic energy that is capable of literally tearing apart the minds of lesser species."
Psionic Storm (or Psi Storm or simply Storm) is an offensive spell cast by the Protoss High Templar. A Psionic Storm that is unleashed will damage any units under it. Psionic Storm has no effect on buildings. Psionic storm damages the Templar when it casts it upon itself. Psionic Storm is not stackable so casting more than one on the same object at the same time does not cause more damage. Psionic Storm does not damage units under a Stasis Field. 
When casting Psionic Storm lightning-like images appear over a 3x3 matrix area. Units in this area rapidly receive 8 volleys of 14 damage. This damage is received regardless of armour or unit size, but since the damage is dealt over a time you can dodge Storms with proper micro, thus sparing your units from receiving all the damage.
In competitive play Storm is often used. Because of its Area of Effect properties, low energy cost and high damage value it is a popular weapon against big late game armies.
Late game PvT often sees the use of Storm as the static Terran army often receives full damage from the spell. Under the cover of the other Protoss ground forces Templars can get close enough to unleash a Storm. When Carriers come into play Storm is particularly effective, albeit mirco-demanding, as the smaller Goliaths clump together moreso than Siege Tanks. In the rare case of a Terran going for a Marine-Medic army, Storm, as well as Reavers, is considered a great counter as the low-HP units quickly die to this spell.
Late game PvP also sees a lot of Storms. With the big armies clashing Storm is a very effective way to damage great parts of the enemy's forces. Storm is also seen as the counter to Reaver play as both the Shuttle and the Reaver are very effectively damaged by a good Storm.
Storm flourishes in the PvZ match-up as most Zerg units are small and have low health. Zerglings, Hydralisks and Mutalisks melt under a well placed storm and the Zerg player is forced to micro away, consequently not attacking. It is common practice to storm Lurker(s) and then sending in one Zealot or Dragoon to finish off the low health lurker(s).
A devastating way in which Storm can be used is the Storm Drop. Storm is very effective versus low-HP units, so it can quickly eliminate workers. A single High Templar can wreak havoc on a mineral line. They are flown in and out of the enemy base using a Shuttle.
In Combination with Other Spells
- Stasised units cannot be damaged by Storm.
- Storm will dispel Hallucinations.
- Storm damage will stack with damage from Irradiate and Plague, but not from another Storm.
- Storm can damage units under Dark Swarm
- Storm will damage units in Maelstrom or Lockdown.
- Defensive Matrix absorbs some damage done by Storm, reducing the total amount of damage to 4. If a Protoss unit is D-Matrixed, this damage will be dealt directly to the HP rather than to the shields first.
- Effort vs. JangBi on Destination - 09/02/22 - SWL 2009, CJ vs. Khan - JangBi using great Storms to obliterate Effort's forces.
- JangBi vs. NaDa on Othello - 08/04/19 - PL 2008, WeMade vs Khan - JangBi's Thunderstorm
- adapted from: StarCraft Compendium: Protoss Strategy <http://classic.battle.net/scc/protoss/special.shtml>