This is extracted from RUNNING.txt file of Apache’s Tomcat 6 distribution. I think it is pretty straight-forward, so there’s nothing else I should add here.

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Running The Apache Tomcat 6.0 Servlet/JSP Container
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Apache Tomcat 6.0 requires the Java 2 Standard Edition Runtime
Environment (JRE) version 5.0 or later.

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Running With JRE 5.0 Or Later
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(1) Download and Install the J2SE Runtime Environment (JRE)

(1.1) Download the Java 2 Standard Edition Runtime Environment (JRE),
release version 5.0 or later, from http://java.sun.com/j2se.

(1.2) Install the JRE according to the instructions included with the
release.

(1.3) Set an environment variable named JRE_HOME to the pathname of
the directory into which you installed the JRE, e.g. c:\jre5.0
or /usr/local/java/jre5.0.

NOTE: You may also use the full JDK rather than just the JRE. In this
case set your JAVA_HOME environment variable to the pathname of
the directory into which you installed the JDK, e.g. c:\j2sdk5.0
or /usr/local/java/j2sdk5.0.

(2) Download and Install the Tomcat Binary Distribution

NOTE:  As an alternative to downloading a binary distribution, you can create
your own from the Tomcat source repository, as described in “BUILDING.txt”.
If you do this, the value to use for “${catalina.home}” will be the “dist”
subdirectory of your source distribution.

(2.1) Download a binary distribution of Tomcat from:

http://tomcat.apache.org

(2.2) Unpack the binary distribution into a convenient location so that the
distribution resides in its own directory (conventionally named
“apache-tomcat-[version]“).  For the purposes of the remainder of this document,
the symbolic name “$CATALINA_HOME” is used to refer to the full
pathname of the release directory.

(3) Start Up Tomcat

(3.1) Tomcat can be started by executing the following commands:
$CATALINA_HOME\bin\startup.bat          (Windows)
$CATALINA_HOME/bin/startup.sh           (Unix)

(3.2) After startup, the default web applications included with Tomcat will be
available by visiting:      http://localhost:8080/

(3.3) Further information about configuring and running Tomcat can be found in
the documentation included here, as well as on the Tomcat web site:      http://tomcat.apache.org

(4) Shut Down Tomcat

(4.1) Tomcat can be shut down by executing the following command:
$CATALINA_HOME\bin\shutdown            (Windows)
$CATALINA_HOME/bin/shutdown.sh         (Unix)