To run a compiled Java program, the machine must have what loaded and running?

The compiled java program is in byte code and Java Virtual Machine(JVM) runs the byte code which is .class file

JRE

JVM

Java Virtual Machine

the JAVA program life cycle.

Plain code -> "compliation" -> bytecode (.class) -> "JVM Running" -> Assembly code.

.class file

jre

JVM (Java Virutal Machine) is mendatory to run compiled Java Program.

A Java Virtual Machine (JVM). 

Note: It depends on the operating system, but the virtual muchine doesn't have to run all the time it might be loaded when a Java application is started (Windows and as an example Eclipse) . Mostly on mobile operating systems at least one JVM instance is running all the time.