What do the different coloured quests mean

There are 4 difficulties: Casual, Normal, Veteran, and Elite, and in addition a hardcore mode. Can be seen on this video at around 2:50.

As the official site states:

Since there are no real benefits for picking hard difficulties other than fun (you don’t get better loot), you are free to play the game how you most enjoy. If you like being overpowered… you can play normal mode. If you like a serious challenge, play elite (which is much harder than very hard in TL 1). If even Elite is not enough for you, you will be able to easily download a difficulty mod and challenge yourself further.

The loot and drop rates don't change, only monster health and damage change.

And there is always the option to create your own difficulties with mods, to suite your specific desires.

I am going to attempt to catalog the phase beast challenges that I am aware of. At the moment, I have a list of 14 challenges. Let me know if you know of any corrections or additions that need to be made here, and I'll add them. I've finally added the challenges added in 1.17.

Act 1 (6 Challenges)

• Stay Close to Raise the Totems - Stand close to each of three totems as they rise out of the ground. Reward: One golden chest for each totem you raise

• Vanquish Your Foes With Haste - Destroy as many enemies as quickly as you can (?). Reward: 1 small chest + 1 medium chest + 2 golden chests

• Poison the Spider Nests - Poison several spider nests while battling the spiders that each one spawns. Reward: Gold + 1 golden chest

• Ignite Braziers in Order - Light the braziers in the correct order, while battling enemies. Reward: Gold + 2 small chests + 1 medium chest

• Protect the Crystals From the Goblin Hordes - Protect five ember crystals from goblin enemies. Reward: 1 small chest for the first crystal + 1 medium chest for each crystal you save (though I have yet to protect all four crystals)

• Navigate the Maze - This challenge was added in version 1.17. Navigate a watery maze, collecting all three mysterious gems if you can. Reward: 3 medium chests + 1 golden chest, which is only available if you collect all 3 mysterious gems

Act 2 (7 Challenges)

• Choose a Door - Battle several waves of monsters, get the dropped key, and pick one of three doors. Reward: One of the following, depending on the door you choose: (a) nothing, (b) medium amount of gold + a few chests, or (c) large amount of gold + a few large chests

• Jackalbeast Gauntlet - Run the gauntlet, collecting gold and fighting Jackalbeasts as you go. Reward: Gold + 1 small chest + 2 medium chests + random items

• Pirate Gauntlet - Run through the ancient pirate ships, collecting gold and battling skeleton pirates along the way. Reward: Gold + 6 small chests + 2 medium chests + random items

• Slime Trio - Defeat the three giant slimes. Reward: Gold + three chests

• Survive in the Arena - Survive the enemies in the arena. Reward: Gold

• Dig for Buried Treasure - This challenge was added in version 1.17. Use shovels to dig for buried treasure among 20 marked places. Reward locations are random, and not all X marks have treasure (some produce a mine, some produce another shovel, and some produce nothing). Reward: 1 big + 3 small treasure chests (only available if you find their hidden locations)

• Avoid the Lava - This challenge has several chests at various height levels. The goal is to get as many chests as you can while staying out of the rising lava.

Act 3 (4 Challenges)

• Defeat the Troll - Defeat the champion-level troll that spawns. Reward: Gold + 2 small chests + 1 medium chest

• Defeat the Witches - Defeat the two champion-level witches. Reward: Gold + 2 medium chests

• Varkolyn Gauntlet - Run the gauntlet, fighting varkolyn enemies and collecting gold along the way. Reward: Gold + 5 small chests + 3 medium chests + random items

• Survive the Attack - Survive an attack by a number of enemies, mines, and robots. Reward: Gold + 3 small chests + 1 medium chest

If there's a fifth challenge in act 3, I have yet to find it.