Groovy is a dynamically typed language, whose syntax is very close to Java, with a number of syntax improvements that allow for lighter code and less boilerplate. It can run through an interpreter as well as being compiled, which makes it good for fast prototyping, scripts, and learning dynamic languages without having to learn a new syntax (assuming you know Java). As of Groovy 2.0, it also has growing support for static compilation. Groovy supports closures and has support for programming in a somewhat functional style, although it's still fairly far from the traditional definition of functional programming.
Clojure is a dialect of Lisp with a few advanced features like Software Transactional Memory. If you like Lisp and would like to use something like it under the JVM, Clojure is for you. It's possibly the most functional language running on the JVM, and certainly the most famous one. Also, it has a stronger emphasis on immutability than other Lisp dialects, which takes it closer to the heart of functional language enthusiasts.
Scala is a fully object oriented language, more so than Java, with one of the most advanced type systems available on non-research languages, and certainly the most advanced type system on the JVM. It also combines many concepts and features of functional languages, without compromising the object orientation, but its compromise on functional language characteristics put off some enthusiasts of the latter.
Groovy has good acceptance and a popular web framework in Grails. It also powers the Gradle build system, which is becoming a popular alternative to Maven. I personally think it is a language with limited utility, particularly as Jython and JRuby start making inroads on the JVM-land, compared to the others.
Clojure, even discounting some very interesting features, has a strong appeal just by being a Lisp dialect on JVM. It might limit its popularity, granted, but I expect it will have loyal community around it for a long time.
Scala can compete directly with Java, and give it a run for its money on almost all aspects. It can't compete in popularity at the moment, of course, and the lack of a strong corporate backing may hinder its acceptance on corporate environments. It's also a much more dynamic language than Java, in the sense of how the language evolves. From the perspective of the language, that's a good thing. From the perspective of users who plan on having thousands of lines of code written in it, not so.
As a final disclosure, I'm very familiar with Scala, and only acquainted with the other two.