Objectives are basically side-quests that you can complete inside your vault that reward anything from caps to lunchboxes and can be as simple as assigning the correct dweller to a room, such as a dweller with high Agility to food production, to as difficult as killing a specific type of creature unarmed. You can check to see what your current objectives, every 24 hours new objectives comes. Objectives can be skipped at the cost of Nuka-Cola Quantum. After you have completed 100, this achievement is yours.