I wrote a couple though some might overlap with what was already said. :p
General quests/achievements:
1. Beat a 2p map solo on x difficulty
2. Beat a 4p map solo x difficulty
3. Beat x map without using any 4 orb cards
4. Do x amount of damage using spells
5. Do x amount of damage using buildings
6. Do x amount of damage using monsters
7. Heal x amount of health to monsters
8. Get the daily reward x ammount of times
9. Beat x amount of enemys with boss afinities
10. Build x amount of orbs
11. Build x amount of power wells
12. Place x amount of a specific card
13. Use monster abilities x amount of times
14. Kill an XL unit using a small unit
15. Destroy x amount of shields
16. Kill x amount of Twilight/Bandit/Lost Soul units
17. Use a revenant doom ability
18. Use a transform ability
19. Gather x amount of power in x map
20. In pvp kill a promo unit using a promo unit
21. In pvp go to t3 before your opponent builds a second monument
22. Do x amount of damage using x unit
23. Absorb x amount of damage using Ashbone Pyro
24. Destroy x amount of buildings using a unit with the siege ability
25. Freeze 100,500,1000 units in matches (could be a nice excuse to give santa clause as an reward :p)
26. Buy 100 cards in the marketplace
27. Sell 100 cards in the marketplace
28. Win a level 1 battlegrounds using only snapjaws
29. Teleport a unit using a tunnel ability
30. Use all Trick magic devices in Oracle correctly on the first time
31. Open x amount of boosters
32. Knock back 100/500/1000 units
33. Have 4 orbs with different elements in a pve match
34. Trade 100 cards
(Map)specific quests/achievements:
1. Switch the gate in Sunbridge shrine x ammount of times
2. Win Guns of Lyr without any damage on the Rogans Stronghold
3. Win the Dwarven Riddle without using the fire altar
4. Win defending Hope without Rogan taking any damage
5. Win the Soultree without Moon health getting under 50%
6. Win Convoy without letting x ammount of walkers die
7. Win Siege of Hope without letting any of the humans die
8. Save all humans in Titans
9. Win the Insane god without letting any of the priest or temples take damage
10. Defeat Mo without any of the banzai birds exploding
11. Win Nightmares end without losing any of the magic nodes
12. Defeat the Red King in under 3 minutes after the battle starts
13. In Blight defeat the ravenheart in under 15 seconds
14. Destroy 10 Raven Scoutships in Raven's end
15. Don't use the spell device more than 10 times in Convoy
16. Win Soultree without letting any of the Lyrish Defenders die
Endgame, (map)specific quests/achievements:
1. Win any 2p expert map solo using the tutorial deck
2. Win a 12p map expert with only one persons on each map and without using any spells
3. Beat Guns of Lyr/Dwarven Riddle on expert solo without building any buildings(including walls)
4. Beat lvl 10 battlegrounds solo without going higher than the second orb
5. Beat all pve maps in under 3 hours
6. Have 100k gold
7. Win 50 pvp matches in a row
8. Destroy all 8 crystals within 3 minutes after destroying the first one in Dwarven Riddle expert solo