I downloaded a tomcat plugin (com.sysdeo.eclipse.tomcat_3.2.1) for Eclispe and put it inside the dropin folder.
When I start my Eclipse, I *do* see the little Tomcat logos, and after setting some values for TOMCAT_HOME, etc, I can start and stop Tomcat server.
I want to see "Tomcat Project" when I click File -> new. I see "Java Project", etc, as usual, but not the "Tomcat Project".
What you are looking for is "Dynamic Web Project" (it's under Web in the Select a Wizard dialog). At one point the Dynamic Web Project wizard will let you choose your server, that's when you hook it up to the Tomcat server you created earlier.