I downloaded Apache Tomcat 6.0.2 And created a new server in Eclipse
The problem is just as drvdijk mentioned, in order to run a webapp on tomcat from eclipse, it needs to be "deployed" to it. This can be done by right clicking the tomcat server -> add and remove
Alternatively, you can try to startup your tomcat server outside of eclipse. Go to your command line and type
$CATALINA_HOME\bin\startup.bat (Windows)
$CATALINA_HOME/bin/startup.sh (Unix)
Where $catalina_home is the directory of where you installed tomcat