Quests in Xenoblade Chronicles: Definitive Edition are the sidequests of the game and mostly optional, for the most part. You will receive them from various residents of the game's world. You will know there is a new quest available from the white "!" over an NPC's head, as well as on the in-game map. You can bring up a list of your quests by either pressing the Down button on the directional pad, or by going into your menu and choosing the Quests option. By using the Y button while on the Quests menu, you can sort them by their region, as well as completed and current (not complete).
There are a good number of quests that will require you to do something in order to make them appear. The majority of these quests will usually be the Affinity with the region. There are five major areas that all NPCs will fall under in the game, and as you do quests in those regions, you will build up Affinity. It is based on a five-star system (with hidden values in between), which can be seen on Affinity Chart in-game. Other prerequisites for quests include specific story progression, having a certain NPC registered on the Affinity Chart, or just another quest.
Note: To make things a little easier on navigation, areas with a lot of quests will be split up into those given by named NPCs, usually Unique Quests (those by named NPCs), Monster Slaying (those that specifically tell you to defeat monsters) and Collection (require gathering items).
Quests fall under a number of categories, with normal ones having nothing denoted on the quest page that pops up. The other kinds are as follows:
Story Quests - Quests under this category are required to continue the story. They are usually marked differently than normal ones, on the quest page in-game, and will have markers a little more prominent.
Timed Quests - A clock is shown on the quest page, meaning that it will disappear after a specific story event has been passed.
Mutually Exclusive Quests - These quests are a problem, as there is no marker for them. Quests that are mutually exclusive mean doing one will make it so the other quest cannot be completed. So, you have a choice on which one to do.
Surprise Quests - These are quests that will appear automatically, once the specific conditions are met. This could be walking at a certain part of the map, or they might be part of another quest.