Graph databases are described as databases that operate over graphs and where relationships between things matter. A graph is a type of structure and the underlying graph of a graph database maps the structure, or schema, of the data stored in the database.

Features:
The term “graph” comes from the use of the word in mathematics. There it’s used to describe a collection of nodes (or vertices), each containing information (properties), and with labeled relationships (or edges) between the nodes.

A social network is a good example of a graph. The people in the network would be the nodes, the attributes of each person (such as name, age, and so on) would be properties, and the lines connecting the people (with labels such as “friend” or “mother” or “supervisor”) would indicate their relationship.

In a conventional database, queries about relationships can take a long time to process. This is because relationships are implemented with foreign keys and queried by joining tables. As any SQL DBA can tell you, performing joins is expensive, especially when you must sort through large numbers of objects—or, worse, when you must join multiple tables to perform the sorts of indirect (e.g. “friend of a friend”) queries that graph databases excel at.

Graph databases work by storing the relationships along with the data. Because related nodes are physically linked in the database, accessing those relationships is as immediate as accessing the data itself. In other words, instead of calculating the relationship as relational databases must do, graph databases simply read the relationship from storage. Satisfying queries is a simple matter of walking, or “traversing,” the graph.

A graph database not only stores the relationships between objects in a native way, making queries about relationships fast and easy, but allows you to include different kinds of objects and different kinds of relationships in the graph. Like other NoSQL databases, a graph database is schema-less. Thus, in terms of performance and flexibility, graph databases hew closer to document databases or key-value stores than they do relational or table-oriented databases.

A typical Graph database example:

How Apache AGE comes into the picture

Apache AGE is PostgreSQL extension that provides graph database functionality.
It aims to bring graph querying abilities to be brought to relational database paradigm.
Users can read and write graph data in nodes and edges. They can also use various algorithms such as variable length and edge traversal when analysing data. And all these graph based data will be internally be handled keeping the relational database

Check Out Apache AGE.
Apache AGE Website
Apache AGE Github