Unpack the tomcat distribution:

unzip jakarta-tomcat-5.0.19.zip

Make sure that 'JAVA_HOME' is set in your environment.

Open up a shell and start tomcat:

On Unix or Mac OS X:

cd jakarta-tomcat-5.0.19/bin
./startup.sh

On Windows

cd jakarta-tomcat-5.0.19/bin
startup

Note: On Unix or Mac OS X you may need to make the scripts executable first:

cd jakarta-tomcat-5.0.19/bin
chmod u+x *.sh

Check to see if it running at http://localhost:8080