A Side-Quest, as its name implies, branches away from the central (or critical-path) goals of a game. It is usually presented to the player as a goal that the player may want to achieve, though it is usually clear that it is not central.
Side-Quests are usually tied in some manner to the central narrative of the game. For example, a Side-Quest may be to perform a certain favour for a well-known character in the game. Thus, they are usually non-repeatable.
Most RPGs feature Side-Quests as a way to provide players with opportunities to diversify the abilities and belongings of the characters they are controlling. For example, in Baldurs Gate, the player is asked to save a lost kitten. If performed, the player is rewarded with gold and treasure that may not be otherwise available to him.
In Elder Scrolls: Oblivion, there are a large number of side-quests available throughout the game. In fact, the majority of the content in the game can be considered a side-quest. Throughout the game, the player consantly runs into quests that definetly not part of the main story line. This is a strong example because these quests can only be done one time and always generate some form of tangible reward for the player. In many cases, some kind of new 'hideout' is the end reward for a long side-quest.
In Grand Theft Auto, the player can commandeer certain vehicles such as an ambulance or a fire truck. When driving these vehicles the player can engage in a side-quest in which he is attempting to fulfill the role of the vehicle he is driving. For example, when driving the fire truck he can rush to locations where there are fires and attempt to extinguish them. The player is rewarded with cash for fulfilling these side quests. In addition, certain upgrades can be earned at certain levels, such as health and armor upgrades.
This is considered a weak example, because the side quest is always available to the player as an option. It can be performed over and over again. Additionally, in the case of GTAIII, they are not integrated into the main narrative of the game.
In Fable for the X-box, the player encounters Traders who desire to be escorted to certain locations. The player receives reward (monetary [and positive noteriety?]) after accepting the mission and after escorting them alive to the desired location. Should the player decide to execute the Trader after accepting the initial accept-mission-reward, the player will receive negative noteriety points and gold.
In both cases, the Trader will eventually respawn in the initial location and the player can repeat either process as many times as they desire.