A network diagram is composed of nodes and lines (usually directional arrows) between them. It is a way of visually organizing the activities. In Project Management they are used to show the activities and the dependencies between them. Other names include "activity network diagram", "activity diagram", or "project schedule network diagram". These diagrams show the activities and the status before and after each activity. The activities can be the arrows -- known as activity on arrow (AOA) -- or on the nodes -- activity-on-node (AON). The status will be on the other element.
Note: Other domains also use network diagrams, but they are not the same. For instance in information technology, they may show the electronic networks used to transmit data between servers and data storage. In other domains, they show social networks.
An activity network diagram is one of the quality management and control tools listed in the PMBOK 5th edition, section 188.8.131.52 as one of the tools and techniques for the PMI process to perform quality assurance.