A quest is request given by a non-player character (NPC) that usually rewards a combination of Experience Points, skill increases, items (armor, weapons, crystals), and coins. Some quests will reward one of these, some will reward nearly all. Most quests are found in towns and cities by talking to the various inhabitants.
Experience rewarded from quests is scaled by level, just as experience from enemies are.
In the first two games, you were notified when you received a new quest but the quest did not appear on-screen when first accepted. In the later games, a popup would show the quest details for you to see before continuing.
Some quests are long, multi-zone quests (such as The Presence (Quest) in Geneforge 5) while others are extremely simple "talk to this person" style quests. Most quests, however, involve killing some sort of rogue or making your way through a deadly zone or area that presents challenges in Mechanics, Leadership, or battle and are confined to those zones.
There is usually a thread of "main storyline" quests that advance and get you to the end of the game and are the only quests that you are required to complete in order to finish the game. The games make no distinction between these and side quests, however, but it is usually very obvious what is considered required. The rest of the quests are optional but tend to be very useful for faster leveling and greater access to training, weapons, and armor.