Apart from hunting and killing your prey to complete quests in Monster Hunter: World you can also capture and benefit from the additional materials and resources you can earn from taking them alive. Capturing your quarry will also unlock that monster to fight against in the game's arena.
In most of the hunt quests you have the choice of whether to kill your prey or take it alive but in some of the optional quests it will be necessary to capture the monster to successfully complete the quest so it is important that you learn how to do this if you want to tick every quest off the list.
Monsters
To capture a monster you will need a trap and some tranquilliser items to immobilise and sedate it completely for transporting. The easiest way to do this is with a Shock Trap and some Tranq Bombs. You can craft a Shock Trap using the Trap Tool and a Thunderbug and the Tranq Bombs by compbining Sleep Herbs with Prashrooms. If you are undertaking an Investigation to specifically capture a creature these items will already be in the Item Box when you appear in the area.
Once you have your trap items you need to track your prey and fight it so that it is critically weakened and trying to escape back to its nest. At this point the monster is vulnerable enough to attempt capturing it so you need to.stop attacking it and instead track it. Place your Shock Trap on the ground and lure the wounded beast to it so the trap shocks it. At this point throw at least two Tranq Bombs at the monster. If it is a large monster you are trying to capture you will need to throw a few more to sedate it so it falls asleep. Once the monster has been captured the quest ends and the beast is transported to Astera.
Pets
If it is an animal you are trying capture to place in your room as a pet you need to equip yourself with a Capture Net. The Capture Net works the same way as the Slingshot, you simply have to aim and shoot it at the animal you are trying to catch. You can only catch one animal to be a pet per mission so you need to think carefully which animal you want to add to your collection.