Well, first I'll say you the types of misions that are in the game. When you do quests, you(or your game partners) can pass out(it's better say die,but it isn't true) only 3 times. If you "die" three times,the quest has been failed. In this game are three types of quests:

Hunting quests: you have to hunt monsters( or capture it, which is also valid)

Capture quests: you have to capture alive the monster, with traps and calming pumps)

Tour quests: you have to wait five minutes and the game will deliver you in the object trunk(blue) a ticket. Then you have to deliver it in the quest trunk (red) and you will finish the quest. That quests are to take objects that you need, and sometimes appears a monster to hunt.

Well, once explained this, I will show you only one mode,because in both appears the same monsters. I'll show you the online mode.In online mode you can play with other players, for example from France,from Italy,from Japan... There you can do the same quests than in the offline mode, but when you finish it, you haven't finished the game. Then in online mode you have the same or very similar quests, but more difficult. You have the quests from 1 to 6 stars, and the quests can be to hunt a monster or more. The most difficult quests are quests from 6 stars and the events.
The events are quests too difficult but apart from that can also be to hunt one or more monsters, in that missions you can obtain rare materials, apart from, materials that in a simple mission of any star you can't get it. Well, before the list of monsters that i'll write in a few moments, I have to write one more something.
From 1 to 3 stars, the quests are the same that in the offline mode (excluding the tour missions or deliver missions to deliver items,because almost all are to deliver the same objects as in offline mode but the double),but quests from 4 to six stars and events aren't in offline mode. I mean. From 1 to three stars, the monsters equally strong that in offline mode, but from 4 to 6 stars and events the monsters are called "+" because they're all twice strong than the offline mode. Each mission has its rules and restrictions, but most have the same rules. Well, finally, the list of the monsters!: