Trolls in The Lord of The Rings
Trolls are humanoid-like creatures created by the first Dark Lord Melkor in mockery of the Ents.
In the Lord of the Rings: The Fellowship of the Ring, a cave troll was found in the mines of Moria. This troll stabbed Frodo who was saved by the mithril armor Bilbo gave him.
The Olog-hai are an advanced breed of trolls created by Sauron. They can withstand sunlight. They are appeared during the latter days of the War of the Ring.
This Olog-hai attacked Aragorn in the Return of the King film. Olog-hai are commonly armed with clubs but this troll is wielding a sword,which implies that he is the troll named Rogash. Rogash allied with the Witch King to conquer Arnor. Rogash is a playable unit in the Battle for Middle Earth II: Rise of the Witch King but he was not present in any of Tolkien's works.
Aragorn and an Olog-hai Troll
Rogash from the Battle for Middle Earth 2: Rise of the Witch King Video Game
Snow trolls live in areas of mountain and ice. They live in the frozen wastelands of Angmar. They didn't exist in any of Tolkien's works because they were created for the Lord of the Rings video games such as the Battle for Middle Earth 2: Rise of the Witch King Expansion Pack and The Lord of the Rings Online: Shadows of Angmar.