JDK (Java Development Kit) is the full toolbox for developers: includes javac compiler, libraries, and JRE. JRE (Java Runtime Environment) is the minimal environment to run Java applications: includes JVM and core libraries. JVM (Java Virtual Machine) is the engine that actually executes the compiled bytecode. Think of it as nested: JDK contains JRE, and JRE contains JVM.
JDK is like a full workshop with tools to build furniture. JRE is like a room with just the assembled furniture. JVM is the person using that furniture.
You are deploying a Java application to a client's production server. Should you install the JDK or the JRE? Justify your choice.