The JRE (Java Runtime Environment) is the equivalent of a JVM but for running applications outside of a browser (like from the DOS prompt). When you run an applet or servlet it is running in a JVM (Java Virtual Machine). It is actually a set of specifications that Sun gives to the makers of browsers to follow when creating their JVMs.
The JRE is a product from Sun which has a JVM in it. A JVM is an implementation of Sun's specifications for something - could be hardware, could be software, could be a browser - which interprets the Java byte code, executing it on a platform.