The native Java API for Neo4j provides a high degree of control over how graphs are stored and queried, allowing developers to optimize queries for the best performance. The Cypher query language is a declarative query language with an innovative SQL-like syntax for graphs, enabling users to tell Neo4j what they want without specifying how to get it. When a Cypher query is executed, Neo4j translates it into an execution plan that describes a set of data operations arranged sequentially until a result is generated. The typical approach involves sending queries and parameters in a POST request to the Neo4j database server using a "driver" framework or library.