Yes, you are thinking of it it the wrong way. The key feature of Nodejs is that it contains, amongst other things, a rather natty embedded HTTP server. Think Jetty. However it might be possible now or in the future to run JavaScript in tomcat.
I recommend going to http://nodejs.org/ and watching the video, it will surely clear the matter up for you.
Meanwhile I would recommend getting hosting which includes SSH access, and preferably root access. Root access will make installing node a whole load easier and allow you to run it on a commonly used port (avoid running node as root, use port mapping) . But you don't need a seperate web server to get started, you can install Node.js on windows, mac and linux.
与恶龙缠斗过久,自身亦成为恶龙;凝视深渊过久,深渊将回以凝视…