Fallout Shelter

After getting 18 dwellers in your Vault, you will unlock the Overseer's Office. This room will allow you to send up to 3 dwellers on Quests.

Some quests have Boss enemies. These do not particularly have to be vital enemies to the quest, but it's advised you search all rooms and kill all enemies in every quest area anyway so you will get this after about 15 quests.

Boss enemies will be similar to normal enemies, only they will do more damage and have a little more health. Some enemies will be unmissable for the Quest you are on, so as long as you're doing quests this will come naturally.

Every Boss enemy has a red skull on them.

This achievement can be done one of two ways or mix of both. You can either send a dweller out to the wasteland and once in a while a map icon with next to it will appear allowing you to explore an are of the wasteland. Or you can do this in quests. To do this in quests it requires you to get 18 dwellers and unlocking the overseer's office. Once you have built it you may begin doing quests. from the quest menu decide which one you want to do based on the required level of dwellers, time to start, and loot available etc. Once you are in the quest you can move around the area similar to how you you would move around in your vault, fighting enemies and collecting loot along the way. Once you get about 90% through the are you will usually come across a little dialogue (Which has to do with the main objective of the quest), once everyone has said their words you will begin a fight with the boss and usually a few other grunts. You can identify the boss based on the red skull next to his or her health bar. After you have kill the boss it will be added to your total value for the achievement. Upon killing your 10th boss, the achievement will pop and is now yours.

If you need further explanation feel free to check out my guide with more detail about this achievement: