Here is the full list of all 16 Bloons TD Battles achievements.
Beat an opponent with a lower battle score
Win your first battle
Win 5 quick play battles in a row
Win 20 sequential quick play battles
Beat an opponent with a higher battle score
Reach 125 battle score
Unlock tier 3 upgrades for all towers
Unlock tier 4 upgrades for all towers
Win a battle using bloons with decals
Get 3 or more decals
Use tower boost and/or bloon boost 100 times
Win 100 quick play battles on assault mode
Win 100 quick play battles on defensive mode
Beat an opponent by using a random loadout
Spawn 100,000 bloons during your battles
Pop 1,000,000 bloons during your battles