The Dragoon is a ranged ground unit that can be warped in at the Protoss Gateway if a Cybernetics Core has been warped in. They are one of the most versatile units in the game, with high health and pushing power, and with proper micro, have the ability to deal high amounts of damage without having to risk any in return due to their high speed and range. Necessary for their superior range is the Singularity Charge upgrade at the Cybernetics Core. Much like Hydralisks, their explosive attack means that they are quite a large deal weaker against small units that can close the distance, like Zerglings and speed-upgraded Zealots, however.
When used with Zealots, the strength of the Dragoon truly shines. Having Zealots fight in close combat, Dragoons can score uncontested hits against enemy units. Dragoons are especially strong at kiting Terran infantry despite their explosive weapon type and Terran's infantry small size (which is what usually forces Terrans into mech against Protoss, on top of that Siege Tanks are quite cost-effective against them).
The Dragoon is the ranged army unit for Protoss, the only one that can really contest other large ground armies (since melee armies find it harder to all attack at once) until Reavers are available with a Robotics Support Bay and then still quite useful and often necessary due to their high burst damage upon being massed and ability to attack air, at least as supplemental defenders.
Dragoon unit AI is, quite famously, amongst the worst of all StarCraft units. However, bug fixes in Patch 1.18 have fixed some of the Dragoon's AI movement bugs. Prior to 1.18, Dragoons would sometimes glitch such that they would not move until the player presses Hold Position or Stop three times. From Patch 1.18 onward, Dragoons should no longer encounter this specific Hold Position / Stop bug. Like many units Dragoons will also stop unnecessarily close to their targets to attack at less than maximum range disconviencing pure-attack-move micro. When their range is upgraded at the Cybernetics Core, Dragoon AI seems to improve slightly. For example, non range-upgraded Dragoons will often run into Spider Mines even when they have vision of them, but range-upgraded Dragoons usually will not.
Oftentimes Dragoons will want to shoot the Zerglings or Zealots closest to them. The way to micro like this is to be running away from the melee units and attack-move in a spot at the ground towards your enemy's units.
Protoss units tend to suffer against Zerg units that are potentially much more damaging for cost. Unlike Terran, no early Protoss composition is quite able to fend off Zerg attacks completely at range while in the open. Hydralisk speed and high explosive damage deal major damage against Dragoons, as well. For this reason primarily Zealot compositions are more popular against Zerg in the early game.
Dragoons are the opening Protoss anti-air, able to stop a single Overlord from scouting reliably, and is commonly created after the first Zealot in the event of the Protoss fast expand. Dragoons in the Zerg matchup have no special utility except being relatively strong at countering Lurkers and fighting off low masses of Mutalisk.
Especially in the Zerg matchup, Dragoons suffer at anti-air as compared against other units. Archons, High Templars with Psionic Storm, and Corsairs are much more effective responses to counter Zerg air.
Dragoons are important in the later game for being effective for preventing Hydralisks from taking free shots at High Templar and Reavers opening with a Gateway-focused army. (Zealots perform this function poorly because they have no ranged attack.)
Towards the late game, the strength of Dragoons drops significantly due to Defilers which can lower their effectiveness heavily with the Defiler's Dark Swarms, as well as the availability of the adrenaline upgrade for Zerglings that allows the Zerg to quickly dispose of unsupported Dragoons. Dragoons, however, are in turn effective against Ultralisks as their explosive damage type allows them to inflict 26 damage per hit (with the weapons +3 upgrade).
Dragoons are the backbone of the Protoss army in early-game PvT, and remain so until Zealot Speed has been researched. Range-upgraded Dragoons outrange Marines and are all but able to ignore Vultures. Therefore, a common micromanagement technique used by Protoss players against an early Terran rush like the FD build is pulling range-upgraded Dragoons barely outside of Marine range, pressing Hold Position to give them a free shot at the outranged Marines, and cycling this manuvuer. This micromanagement technique is known as Dragoon dancing. A Dragoon with the range upgrade can even hit a Bunker without being hit itself by unupgraded Marines, if a player inches the Dragoon to the end of its range. A stimmed Marine dies in three Dragoon shots, although it is rare to see Terran not teching straight to Siege Tanks.
While in the early game Dragoons are entirely sufficient for holding off smaller pushes, but later need to be supported by Zealots. (For the precise technique, see Push Breaking.) Early Dragoons often take advantage of Terran not having yet produced sufficient mech.
Dragoons fare poorly against Siege Tanks sieged up, dying in three hits, far before they are able to get shots in on a siege line built at the same use of the economy but are still useful for keeping them from pushing out directly due to their speed. Even against superior ground-based armies Dragoons are often used in an army for this purpose, although Zealots are the ones that more often than not dish out the majority of punishment toward sieging and unsieging tanks, if the Protoss chooses to go on the offensive. They also can kite Goliaths in a pinch, although the units they would be defending in such a manuever (Corsairs and Carriers) are easy prey for Goliaths and they probably would have to fight directly.
They serve this function until the late-game until Arbiters and other utility units allow Protoss to bypass or destroy defensive lines and raid bases more directly.
Dragoons compose the majority of early-game armies in PvP because they are ranged and faster than Zealots (before the Zealot's Speed Upgrade is researched), with proper micromanagement Dragoons can kite Zealots indefinitely. In late-game PvP, Dragoons can support armies based on the engaging power of Speedlots, High Templar, and Archons. In PvP Dragoons are also essential for defending against Reaver drops and other Shuttle play. If a Protoss player anticipates a drop, he will often keep three Dragoons in his main to repel it.
Tempest vs. Light - 2009-01-03
VOD: Game from the 16th Survivor tournament. Light starts off with a BBS in the middle of the map. Tempest is caught off-guard but proceeds to defend with a single Dragoon, supported by a Shield Battery. Microing that single dragoon back and forth in a tug-a-war match between Light's endless stream of marines, Tempest is able to repel the push and to win the game. The Dragoon ended up with 33 kills in total.