What situations trigger the Master Sword’s special power

They're always in the same spot (and after a Blood Moon, they all respawn). You can see a full map of them at IGN's Hyrule map.

You can find a few types of bosses:

• Hinox: These are the one eyed giants. They primarily drop a handful of weapons.
  • Stalnox: These are like Hinox, but only available to fight at night. They're a bit different from Hinox as they throw bones at you frequently (making it harder to run away and fight them at the same time). You can stop them by shooting them in the eye.
• Talus: These are the rock type monsters. They drop lots of valuable gems.
  • Igneo Talus: Same as a Talus, but made of hot rock that must be cooled down with ice before climbing him. A full 2*+ Fireproof set also works because it negates burning.
  • Frost Talus: Ice counterpart to the Igneo Talus. The armor that helps here is the Rito set or Zant's Helmet.
  • Igneo Talus Titan: A very large Igneo Talus that's encountered during the Ballad of Champions. Too large to be climbed normally, but updrafts spawn around it.
• Molduga: 4 of these can be found in Gerudo Desert. They drop treasure chests and crafting materials.
  • Moldoqueen: Another Ballad of Champions enemy. Didn't notice any major difference to a normal Molduga, except for buffed health.

A worthwhile mention is Lynels. They don't get a health bar, but they're stronger than most bosses and drop some crafting materials for end-game stuff as well. They're also the only enemy that grows stronger as you defeat Titans, rather than through the standard difficulty scaling system (or not at all for bosses).

It takes 10 real time minutes for the Master Sword to recharge after "running out of charge". This cannot be sped up.

Bigger spoilers ahead:

Note that when fighting Guardians, Divine Beast bosses, and Ganon, the blade will glow Blue and the damage will double to 60. While in this state, it has greatly increased durability and can be used for a great amount of time, as long as it hitting the above enemies. Hitting rocks etc while the blade is glowing blue will still lose durability at a normal rate.