JVM (Java Virtual Machine) is an abstract machine that enables your computer to run a Java program.
When you run the Java program, Java compiler first compiles your Java code to bytecode. Then, the JVM translates bytecode into native machine code (set of instructions that a computer's CPU executes directly).
Java is a platform-independent language. It's because when you write Java code, it's ultimately written for JVM but not your physical machine (computer). Since, JVM executes the Java bytecode which is platform independent, Java is platform-independent.
If you are interested in learning about JVM Architecture, visit The JVM Architecture Explained.
JRE (Java Runtime Environment) is a software package that provides Java class libraries, along with Java Virtual Machine (JVM), and other components to run applications written in Java programming. JRE is the superset of JVM.
If you need to run Java programs, but not develop them, JRE is what you need. You can download JRE from Java SE Runtime Environment 8 Downloads page.
JDK (Java Development Kit) is a software development kit to develop applications in Java. When you download JDK, JRE is also downloaded, and don't need to download it separately. In addition to JRE, JDK also contains number of development tools (compilers, JavaDoc, Java Debugger etc).
If you want to develop Java applications, download JDK.
Here's the relationship between JVM, JRE, and JDK.