JVM
The Java Virtual Machine (JVM) is the virtual machine that runs the Java bytecodes. The JVM doesn't understand Java source code; that's why you need compile your *.java files to obtain *.class files that contain the bytecodes understood by the JVM. It's also the entity that allows Java to be a "portable language" (write once, run anywhere). Indeed, there are specific implementations of the JVM for different systems (Windows, Linux, macOS, see the Wikipedia list), the aim is that with the same bytecodes they all give the same results.
JDK and JRE
To explain the difference between JDK and JRE, the best is to read the Oracle documentation and consult the diagram:
Java Runtime Environment (JRE)
The Java Runtime Environment (JRE) provides the libraries, the Java Virtual Machine, and other components to run applets and applications written in the Java programming language. In addition, two key deployment technologies are part of the JRE: Java Plug-in, which enables applets to run in popular browsers; and Java Web Start, which deploys standalone applications over a network. It is also the foundation for the technologies in the Java 2 Platform, Enterprise Edition (J2EE) for enterprise software development and deployment. The JRE does not contain tools and utilities such as compilers or debuggers for developing applets and applications.
Java Development Kit (JDK)
The JDK is a superset of the JRE, and contains everything that is in the JRE, plus tools such as the compilers and debuggers necessary for developing applets and applications.
Note that Oracle is not the only one to provide JDKs.
OpenJDK
OpenJDK is an open-source implementation of the JDK and the base for the Oracle JDK. There is almost no difference between the Oracle JDK and the OpenJDK.
The differences are stated in this blog:
Q: What is the difference between the source code found in the OpenJDK repository, and the code you use to build the Oracle JDK?
A: It is very close - our build process for Oracle JDK releases builds on OpenJDK 7 by adding just a couple of pieces, like the deployment code, which includes Oracle's implementation of the Java Plugin and Java WebStart, as well as some closed source third party components like a graphics rasterizer, some open source third party components, like Rhino, and a few bits and pieces here and there, like additional documentation or third party fonts. Moving forward, our intent is to open source all pieces of the Oracle JDK except those that we consider commercial features such as JRockit Mission Control (not yet available in Oracle JDK), and replace encumbered third party components with open source alternatives to achieve closer parity between the code bases.
Update for JDK 11
An article from Donald Smith try to disambiguate the difference between Oracle JDK and Oracle's OpenJDK : https://blogs.oracle.com/java-platform-group/oracle-jdk-releases-for-java-11-and-later
As mentionned in comments by @Alan Evangelista, Java Web Start has been deprecated by Oracle in Java SE 9 and removed in Java SE 11.
