Mission Quests are the quest you will spend your time doing and are the only quests that give rewards aside from Reputation. These grant XP, Dram, and Reputation. Mission quests will show up in the missions tab in the inventory section of the player menu. Each NPC has a number of Mission Quests that it will give, available depending on the maximum class level of the player. These currently go from level 1 to 30 leveled quests.
Weekly quests are randomly picked from the quest pool of that NPC and given out on a weekly basis for a full reward.
Daily quests are obtained by selecting the Plus icon in the same mission menu when Mission Credits are available. Mission Credits are provided at a rate of 5/day for a maximum of 25, starting the day after a player first joins the game and each Credit can be spent to obtain one random mission from the chosen vendor that is below the current class level, has been completed before, and is not currently in the player's Missions list.
All Mission Quests can be turned in for Bronze, Silver, or Gold rewards and provide increasing Reputation and XP as the reward tier increases (200 Reputation + 12000 base XP for Gold) - note that this XP is subject to the 10% bonus for Enabled World PVP.
Find out here how to use Mission Quests to get the most experience.