One way is to use Eclipse to build a new WAR, then go to the Tomcat admin webapp and tell Tomcat to upload and deploy that WAR.
It's possible that the "Run on Server" menu option might also be capable of deploying a rebuilt WAR, but I've never tried it.
Mostly when
testing I build a Tomcat context XML file and store it in TOMCAT_HOME/conf/Catalina/localhost/ where the codebase for the webapp context is my build directory in the Eclipse project. That way I don't have to build and deploy a WAR -- just do whatever it takes to modify the build directory of the Eclipse project. That's convenient when modifying classes and
JSP files, since Tomcat will normally pick up the changes after a minute or 2.
For some things - mostly config file changes and changes to the init code to servlets, a Tomcat restart is required, which the sysdeo plugin makes easy, since all I have to do is click the litle cat buttons on the Eclipse toolbar.