Quests is simply a term for directed game play. During your playtime in UO you will find some NPCs that require your assistance, sometimes they will reward you with magical items or gold.

The quest engine allows for hundreds of new quest options, including the opportunity to engage in multiple quests at the same time. Some quests are very simple in nature - seek out a certain monster and kill it, for example. Others require you to explore new dungeons or areas of mystery. Still others involve crafting, and reward you with new runic kits or crafting recipes. You can even change your race through questing. The quest opportunities are endless and there is something for everyone.

Quest Types:
# Kill: Specifies a type, quantity, and in some instances, location of creatures you must kill.
# Obtain: Requires you to gather specific items and return them to the Quest Giver. Some items come from monsters that must be slain, some items must be crafted, and other items must be found elsewhere. Some of these quests require you to gather and return a certain quantity of a specific item.
# Escort: Help an NPC travel to a given location.
# Delivery: Take a quest item from one NPC and deliver it to another NPC.

Rewards:
Rewards from each quest vary a lot, but completing quests that give either Large bags of Treasure or Strongboxes are the only way to get Special Pigments (dyes that allow you to dye armors, weapons and clothings).

For a complete list of quests visit (almost all those quests are available on UO: Depths, but there are also some new quests)
http://www.uoguide.com/Quests_by_Area
http://www.uoguide.com/Quests_by_NPC