A task dependency is a link between two tasks that typically signifies that the completion of one task depends on the completion of the second.  Tasks can have dependencies to/from multiple tasks.

Once linked, the tasks will be marked with a chain-link icon in the task list.  In the TASK INFO view, the task name of the linked task(s) will be shown next to an arrow that indicates the direction of the dependency.  To switch to the linked task, simply click on the task name from the Dependency list.

Task dependencies are also depicted in the SCHEDULE view as an arrow from the prerequisite task to the dependent task (assuming both have schedules specified).

blue arrow indicates that the two tasks are in sequential order. 

A red arrow indicates that there's a scheduling conflict because the first task is expected to be completed only after the second is due to begin.  Whenever this happens project admins will also be notified in their daily digest email of any "broken task dependencies" in the project schedule.  With the task owner they can decide how to resolve the conflict, for example by delaying the start of the second task or restructuring the tasks to minimize delays caused by the dependency.