I see here that there are a load of languages aside from Java that run on the JVM. I\'m a bit confused about the whole concept of other languages running in the JVM. So:
Because the JSR process is rendering Java more and more dead: http://www.infoq.com/news/2009/01/java7-updated
It's a shame that even essential and long known additions like Closures are not added just because the members cannot agree on an implementation.
The .NET languages are more for show than actual usefulness. Each language has been so butchered, that they're all C# with a new face.
There are a variety of reasons to provide alternative languages for the Java VM:
What the JVM can do is defined by the JVM's bytecode (what you find in .class files) rather than the source language. So changing the high level source code language isn't going to have a substantial impact on the available functionality.
As for what is required to write a compiler for the JVM, all you really need to do is generate correct bytecode / .class files. How you write/compile code with an alternate compiler sort of depends on the compiler in question, but once the compiler outputs .class files, running them is no different than running the .class files generated by javac.
The advantage for these other languages is that they get relatively easy access to lots of java libraries.
The advantage for Java people varies depending on language -- each has a story tell Java coders about what they do better. Some will stress how they can be used to add dynamic scripting to JVM-based apps, others will just talk about how their language is easier to use, has a better syntax, or so forth.
What's required are the same things to write any other language compiler: parsing to an AST, then transforming that to instructions for the target architecture (byte code) and storing it in the right format (.class files).
From the users' perspective, you just write code and run the compiler binaries, and out comes .class files you can mix in with those your java compiler produces.
You need other languages on the JVM for the same reason you need multiple programming languages in general: Different languages are better as solving different problems ... static typing vs. dynamic typing, strict vs. lazy ... Declarative, Imperative, Object Oriented ... etc.
In general, writing a "compiler" for another language to run on the JVM (or on the .Net CLR) is essentially a matter of compiling that language into java bytecode (or in the case of .Net, IL) instead of to assembly/machine language.
That said, a lot of the extra languages that are being written for JVM aren't compiled, but rather interpreted scripting languages...
The reason is that the JVM platform offers a lot of advantages.
The languages Sun is trying to support with their Scripting spec (e.g. Python, Ruby) are up and comers largely due to their perceived productivity enhancements. Running Jython allows you to, in theory, be more productive, and leverage the capabilities of Python to solve a problem more suited to Python, but still be able to integrate, on a runtime level, with your existing codebase. The classic implementations of Python and Ruby effect the same ability for C libraries.
Additionally, it's often easier to express some things in a dynamic language than in Java. If this is the case, you can go the other way; consume Python/Ruby libraries from Java.
There's a performance hit, but many are willing to accept that in exchange for a less verbose, clearer codebase.