In Monster hunter you'll be sent doing all kinds of Quests. These main ones are below, You can accept these quests from the your Handler at the Quest Counter or Quest Board.
Usually the quests will come with conditions that must be completed for you to finish the Quest. Majority of them will be to Defeat monsters, Capture Monsters, delivering items or stopping a monster's advance.
Lets talk about some of these quests in categories.
Assigned:
This is the assigned quests that's tied directly to the story's progression. Doing these quests will push the story forward and unlock more quests, items, food and more for you. it's important that you do these. Note that you cannot repeat these missions once completed.
Optional:
These quests have nothing to do with the story's progression but they can be good for hunting and farming specific materials and resources. These quests can be repeated as many times as you want.
Investigation:
These are randomed quests that you can accept. Once player has registered the quest then they will show up in the "post a new quest" menu. These quests have a number of attempts, That's now many times you can complete these quests. The nice things about these quests is that they give up to 3 rewards, in the tier of Bronze, Silver and Gold. Obviously gold rewards usually are the best.
Event:
These quests are Event quests that are released by the developers for special occasions. Currently there is a PS4 Exclusive Collaboration event with Horizon Zero Dawn. You can read about it here. Usually these quests offer limited time only rewards. So make sure to do them while they're around!