Answering your short question:
In JVM thread pools are abstracted behind java.util.concurrent.ExecutorService interface. There are different implementations of this interface but in most cases methods of this interface would suffice.
To create specific thread pool, take a look at java.util.concurrent.Executors class:
http://docs.oracle.com/javase/6/docs/api/java/util/concurrent/Executors.html
which contains static factory methods for creating different implementations of ExecutorService interface. You may be interested in newFixedThreadPool(int threadsNumber) and newCachedThreadPool methods.
For more general information on Executors in JVM you may want to read following Oracle's tutorial: http://docs.oracle.com/javase/tutorial/essential/concurrency/executors.html
So, to use thread pool (ExecutorService) under Tomcat you should do the following:
.1. Create and register in web.xml instance of javax.servlet.ServletContextListener interface (which would act like an entry point to your webapplication) if it's not done yet.
.2. In contextInitialized(ServletContextEvent) method, you create instance of ExecutorService (thread pool) and store it in ServletContext attribute map, so that it can be accessed from any point in you webapp e.g.:
// following method is invoked one time, when you web application starts (is deployed)
@Override
public void contextInitialized(ServletContextEvent servletContextEvent) {
    // ...
    final int numberOfThreads = ...;
    final ExecutorService threadPool = Executors.newFixedThreadPool(numberOfThreads); // starts thread pool
    final ServletContext servletContext = servletContextEvent.getServletContext();
    servletContext.setAttribute("threadPoolAlias", threadPool);
    // ...
}
// following method is invoked one time when your web application stops (is undeployed)
public void contextDestroyed(ServletContextEvent servletContextEvent) {
    // following code is just to free resources occupied by thread pool when web application is undeployed
    final ExecutorService threadPool = (ExecutorService) servletContextEvent.getServletContext().getAttribute("threadPoolAlias");
    threadPool.shutdown();
}
.3.  Somewhere in Servlet.service method or anywhere in your webapp (you should be able to obtain reference to ServletContext almost anywhere from webapp):
Callable<ResultOfMyTask> callable = new Callable<ResultOfMyTask>() {
    public ResultOfMyTask call() {
        // here goes your task code which is to be invoked by thread pool 
    }
};
final ServletContext servletContext = ...;
final ExecutorService threadPool = (ExecutorService) servletContext.getAttribute("threadPoolAlias");
final Future<ResultOfMyTask> myTask = threadPool.submit(callable);;
You should store reference to myTask and can query it from other threads to find out whether it's finished and what is the result.
Hope this helps...