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:
The advantage to having other languages for the JVM is quite the same as the advantage to having other languages for computer in general: while all turing-complete languages can technically accomplish the same tasks, some languages make some tasks easier than others while other languages make other tasks easier. Since the JVM is something we already have the ability to run on all (well, nearly all) computers, and a lot of computers, in fact already have it, we can get the "write once, run anywhere" benefit, but without requiring that one uses Java.
Writing a language/compiler for the JVM isn't really different from writing one for a real machine. The real difference is that you have to compile to the JVM's bytecode instead of to the machine's executable code, but that's really a minor difference in the grand scheme of things.
Writing code for a language other than Java in the JVM really isn't different from writing Java except, of course, that you'll be using a different language. You'll compile using the compiler that somebody writes for it (again, not much different from a C compiler, fundamentally, and pretty much not different at all from a Java compiler), and you'll end up being able to run it just like you would compiled Java code since once it's in bytecode, the JVM can't tell what language it came from.