Which is the best application server and the best ide for an absolute beginner for locally hosting servlets
One good solution is Eclipse + Tomcat.
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Eclipse and netbeans each have a distribution that includes all you need for Servlet development, including a server.
Between the two, eclipse is the more powerful, but for a beginner it could be a bit overwhelming. Netbeans is less configurable, but that also means there's less to distract you from the "normal" way of doing things, which is probably the best for a beginner.
I recommend using Eclipse JEE version, together with JBoss Application Server. You can download Eclipse plugin: JBoss Tools which adds alot of nice plugins to Eclipse.
You can download JBoss AS (Application Server) for free. All products are free and opensource.
I'd recommend NetBeans + Apache Tomcat/Jetty is you're just looking into servlet development. If you plan to do some EJB3/3.1 development as well have a look at Glassfish or JBoss AS.
NetBeans integrates great with all of these and requires far less configuration than Eclipse to get you started.
You might consider as an IDE option IntelliJ IDEA as well, it's heavier that Eclipse and NetBeans, but it has some compelling features of its own...