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:
Different languages are tailored to different tasks. While certain problem domains fit the Java language perfectly, some are much easier to express in alternative languages. Also, for a user accustomed to Ruby, Python, etc, the ability to generate Java bytecode and take advantage of the JDK classes and JIT compiler has obvious benefits.
Turning this on its head, consider you want to design a new language and you want it to run in a managed runtime with a JIT and GC. Then consider that you could:
(a) write you own managed runtime (VM) and tackle all sorts of technically difficult issues that will doubtless lead to many bugs, bad performance, improper threading and a great deal of portability effort
or
(b) compile your language into bytecode that can run on the Java VM which is already quite mature, fast and supported on a number of platforms (sometimes with more than one choice of vendor impementation).
Given that the JavaVM bytecode is not tied so closely to the Java language as to unduly restrict the type of language you can implement, it has been a popular target environment for languages that want to run in a VM.
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.
Java is a fairly verbose programming language that is getting outdated very quickly with all of the new fancy languages/frameworks coming out in the past 5 years. To support all the fancy syntax that people want in a language AND preserve backwards compatibility it makes more sense to add more languages to the runtime.
Another benefit is it lets you run some web frameworks written in Ruby ala JRuby (aka Rails), or Grails(Groovy on Railys essentially), etc. on a proven hosting platform that likely already is in production at many companies, rather than having to using that not nearly as tried and tested Ruby hosting environments.
To compile the other languages you are just converting to Java byte code.
It's much easier for a compiler writer to generate JVM or CLR byte-codes. They are a much cleaner and higher level abstraction than any machine language. Because of this, it is much more feasible to experiment with creating new languages than ever before, because all you have to do is target one of these VM architectures and you will have a set of tools and libraries already available for your language. They let language designers focus more on the language than all the necessary support infrastructure.
Answering just your second question:
The JVM is just an abstract machine and execution model. So targetting it with a compiler is just the same as any other machine and execution model that a compiler might target, be it implemented in hardware (x86, CELL, etc) or software (parrot, .NET). The JVM is fairly simple, so its actually a fairly easy target for compilers. Also, implementations tend to have pretty good JIT compilers (to deal with the lousy code that javac produces), so you can get good performance without having to worry about a lot of optimizations.
A couple of caveats apply. First, the JVM directly embodies java's module and inheritance system, so trying to do anything else (multiple inheritance, multiple dispatch) is likely to be tricky and require convoluted code. Second, JVMs are optimized to deal with the kind of bytecode that javac produces. Producing bytecode that is very different from this is likely to get into odd corners of the JIT compiler/JVM which will likely be inefficient at best (at worst, they can crash the JVM or at least give spurious VirtualMachineError exceptions).