Groovy is a programming language that compiles to byte codes that are interpreted by the Java virtual machine. It's the evolutionary next step for Java -- it vastly simplifies Java coding, and goes beyond standard Java to things Java doesn't normally do. Of all the non-Java languages that work on the JVM, Groovy is the easiest to learn and has the tightest integration with existing Java classes.
The message board this week is full of examples of when and where to use Groovy.
If you want to give it a try, I think using it for automated testing is a great idea because it's not production code so there's less pressure while you're still learning, and you get useful test showing your code works as expected. If you use Spock (a Groovy based testing framework compatible with JUnit) then you can do some Behavior Driven Development which can be used to show the business folks that their wishes are being implemented too.
SCJP, SCJD, SCEA 5 "Any sufficiently analyzed magic is indistinguishable from science!" Agatha Heterodyne (Girl Genius)