Predecessors (and their counterparts, successors) are how you as the project manager tell Project what the logical flow of tasks should be throughout your project. This tab of the Task Information dialog allows you to see all the tasks that are predecessors for the task currently being edited in the dialog. This section shows you how to create a new predecessor.
As in all tabs in this dialog, you can edit the task Name and Duration fields.
The grid on this tab lists all the tasks that are predecessors to the task named in the Name field.
The type of link is shown in the Type field. In this case, Define Requirements has a Finish-to-Start link to this task, meaning that the Define Requirements task must finish before the Write Specifications task can start.
The Lag field lets you define a span of days between the predecessor side of the link and the successor side of the link. This means that if you add a one-day lag on the link shown, Write Specifications cannot start until one full day after Define Requirements finishes.