windows web hosting in own server

Register for a domain name somewhere.

I have a static IP , So is is it require to register a DNS name from the out side provider .

Do you want anyone who wishes to access the site to have to type in your ip address (is this even reachable from the rest of the world) or do you want them to be able to use a name, i.e. www.doit.com). If they should be able to use a name then register for a domain. If you wish to be accessed through an internet provider (i.e. www.aol.de/yourinfo) then talk to them. This is a development forum, not an icann info desk.

also make sure your ISP actually allows you to host a server on your connection.
Many apparently do not (mine does, if they didn't they'd be looking rather silly as they provide me with server to server mail delivery among other things).

Do make sure you have everything properly secured though. Leaky server software (especially poorly configured databases and poorly shielded database access) are a major source of intrusions.

Instead of Tomcat I'd advise an industrial strength application server (though Tomcat isn't bad, there's a lot better) like Sun Java System Application Server 8 (or GlassFish, its successor) which are free for download from Sun.
http://java.sun.com/javaee/downloads/index.jsp

At work we're running several dozen servers on JBoss, also very nice.