The Frozen, Static and Fiery shrine play a key role in the Legendary Quest. To create the orbs, you require either a Thunder Stone, Water Stone, or Fire Stone. A normal Orb (obtained by boss pokemon) is needed to create the Elemental Orb and to spawn the legendary you must place the Orb inside the shrine at the biome the bird spawns at, only when the orb is full. For example the Fiery Orb you must place that inside the Moltres shrine in the Extreme Hills biome.
To fill the orb up, you need to defeat 375 or more wild pokemon of any type (This is a tested bug, the first orb in your inventory will be filled by defeating any wild Pokemon). When it is filled, you have to put the orb in the shrine to summon the the legendary bird.
The legendary bird shrines are easier to find corresponding to their biome. Frozen Shrines are naturally found in taigas, Ice plains, and Ice Mountains. Static Shrines usually are found in Savannas or Savanna Plateaus. Fiery Shrines are found in Mesa. Each shrine is extremely difficult to find.