Quests are missions for the player to participate in, and very often the successful completion of one these missions leads to a new task. When finishing one quest leads to another, players refer to that as a "chain." There is no way to know for certain if a quest is a chain or not, although generally quests that do not lead to a follow-up will offer items as a reward and "links" in the chain will usually just offer money.
Usually, quests are direct consequences from the same person. For example, Kobold Camp Cleanup leads to Investigate Echo Ridge, which then leads to Skirmish at Echo Ridge. Each of these quests comes from the same questgiver. As soon as the first quest is completed, Marshal McBride will immediately offer the player the second quest.
Occasionally, the quest will come from a different source. This is fairly uncommon except when the new questgiver is right next to the old one. The God Hakkar is succeeded by http://wow.allakhazam.com/db/quest.html?wquest=5065 The Lost Tablets of Mosh'aru], from a very different questgiver in a very different place. There are also times when completing a quest leads to two or more new quests, or that the new quest will not be available to the player because (usually) they are too low in level.
Quests that are later legs of a chain cannot be shared.