This week's book giveaway is in the OO, Patterns, UML and Refactoring forum. We're giving away four copies of Refactoring for Software Design Smells: Managing Technical Debt and have Girish Suryanarayana, Ganesh Samarthyam & Tushar Sharma on-line! See this thread for details.
The most popular use case for Groovy is probably web development, using the Grails framework. Also, it works very well as a "scripting" language, and a lot of people are using it to write JUnit tests for their Java code.
I would recommend checking out the "Groovy In Action" book if you would like a general overview of the language. Otherwise, Google is your friend.
Groovy can also be very useful in an SOA environment that passes XML messages between services. Groovy's XmlSlurper and MarkupBuilder classes make it so much incredibly easier to work with XML than plain Java!
The way I started programming Groovy was by writing tests using it. This is easy because you only have to add the Groovy jar to your project classpath, and you can just write a new test that extends GroovyTestCase. If you don't know how to write the code you want in Groovy, just write it in Java right there in the Groovy class.