you don't need anything you don't have on your LAN connection, but you may need to open some ports on your firewall (especially the port on which Tomcat is serving content over http, usually port 8080 with Tomcat though using port 80 is nicer as it's the standard for http).

Domain names you'll need to buy, there are many companies selling them over the net for not a lot of money per year.
Make sure your ISP allows you to do this though, some (many?) will kick you off the network if you host a server over your home connection.
Others will allow it if you upgrade to a corporate account.
Some also sell (or help you to buy) domain names and hosting services.