Below is a simple ColdFusion application architecture using Apache as web server and Tomcat as JEE application server.

1. Request workflow

Apache: In the Apache configuration file we have a virtual host <VirtualHost host1:80> with DocumentRoot /var/www/theColdFusionApp
Apache receives the request – example: http://123.45.67.89:80.
Hosts file: In /etc/hosts the host host1 is associated with the IP: 123.45.67.89 host1
AJP module: If there is a non-CF request, Apache serves the request itself (ex. html, css, js, image). If there is a CFM request, the AJP module forwards it to Tomcat: ProxyPassMatch ^/(.*\.(cf[cm]|cfml))$ ajp://host1:8080/$1 (Apache config file)
Tomcat has an AJP connector on that port: <Connector port="8080" protocol="AJP/1.3"/>
and an engine to process the requests:

<Engine name="Catalina" defaultHost="host1">
 <Host name="host1"  appBase="webapps">
    <Context path="" docBase="/var/www/theColdFusionApp/"/>
 </Host>

Both entries above are in the Tomcat server.xml config file.
Application folders: In /var/www/theColdFusionApp we have both the CF files (WEB-INF and CFIDE folders) and the actual application folders. More details below.

2. Folder structure and configuration files

/var/www/theColdFusionApp
  /CFIDE/
      /administrator/ 
  /WEB-INF/
      /cfusion/lib/neo-*.xml
      /web.xml
  /META-INF/
  /CSS/
  /JS/
  /IMG/
  *.cfm
/opt/tomcat/
  /conf/
     /server.xml
  /logs/
/etc/httpd/conf.d/host1.conf
/etc/hosts

Line 5: the WEB-INF/cfusion/lib folder contains the ColdFusion configuration files. Basically this is where ColdFusion saves its settings when you make modifications in CF Admin. Some of the important files are neo-datasource.xml and neo-runtime.xml (contains the custom tag locations, the mappings, cfx extensions, etc)
Line 6: the file WEB-INF/web.xml contains some configuration settings shared between CF and Tomcat. One of the most important is the cf.class.path
Line 14: server.xml is the main Tomcat configuration file

3. Troubleshooting

Permissions: make sure that the scripts in /opt/tomcat/bin/ are executable (chmod +x *.sh)
Compatibility: make sure that the ColdFusion version is compatible with Tomcat and Java versions (java -version). For instance, CF11 is not working with Tomcat 8 at the moment.
Clean up Tomcat folder: if Tomcat fails to start or times out, make sure that you clean up the following Tomcat folders: /opt/tomcat/webapp, /opt/tomcat/work/Catalina, /opt/tomcat/conf/Catalina
Restart: in my case it happened that the Tomcat startup process hung up at the step Starting security.... I didn’t find the exact cause for this, but the workaround was to restart the machine.
Log files: You should watch the following log files:

/opt/tomcat/logs/catalina.out
/var/www/theColdFusionApp/WEB-INF/cfusion/logs/*
/var/log/httpd/*

A clean Tomcat startup should look in catalina.out like below. You will know when things are up and running when you’ll see the message ColdFusion: application services are now available (line 50 below) 🙂

Good luck!

Mar 19, 2015 1:03:08 PM org.apache.coyote.AbstractProtocol init
INFO: Initializing ProtocolHandler ["ajp-bio-8101"]
Mar 19, 2015 1:03:08 PM org.apache.catalina.startup.Catalina load
INFO: Initialization processed in 9864 ms
Mar 19, 2015 1:03:08 PM org.apache.catalina.core.StandardService startInternal
INFO: Starting service Catalina
Mar 19, 2015 1:03:08 PM org.apache.catalina.core.StandardEngine startInternal
INFO: Starting Servlet Engine: Apache Tomcat/7.0.59
Mar 19, 2015 1:03:18 PM org.apache.catalina.util.SessionIdGeneratorBase createSecureRandom
INFO: Creation of TestRandom instance for session ID generation using [SHA1PRNG] took [277] milliseconds.
03/19 13:03:41 INFO License Service: Flex 1.5 CF Edition enabled
03/19 13:03:41 INFO Starting Flex 1.5 CF Edition
Mar 19, 2015 1:03:42 PM org.apache.catalina.core.ApplicationContext log
INFO: ColdFusionStartUpServlet: ColdFusion: Starting application services
Mar 19, 2015 1:03:42 PM org.apache.catalina.core.ApplicationContext log
INFO: ColdFusionStartUpServlet: ColdFusion: VM version = 25.31-b07
Mar 19, 2015 13:03:43 PM Information [secure-startStop-1] - Starting logging...
Mar 19, 2015 13:03:43 PM Information [secure-startStop-1] - Starting license...
Mar 19, 2015 13:03:44 PM Information [secure-startStop-1] - Evaluation Edition enabled
Mar 19, 2015 13:03:44 PM Information [secure-startStop-1] - Starting crypto...
Mar 19, 2015 13:03:45 PM Information [secure-startStop-1] - Installed JSafe JCE provider: Version 6.0 Crypto-J 6.0, EMC Corporation. 
Mar 19, 2015 13:03:45 PM Information [secure-startStop-1] - Starting security...
Mar 19, 2015 13:03:48 PM Information [secure-startStop-1] - Starting scheduler...
Mar 19, 2015 13:03:48 PM Information [secure-startStop-1] - Starting WatchService...
Mar 19, 2015 13:03:48 PM Information [secure-startStop-1] - Starting debugging...
Mar 19, 2015 13:03:48 PM Information [secure-startStop-1] - Starting sql...
Mar 19, 2015 13:03:49 PM Information [secure-startStop-1] - Pool Manager Started
Mar 19, 2015 13:03:49 PM Information [secure-startStop-1] - Starting mail...
Mar 19, 2015 13:03:49 PM Information [secure-startStop-1] - Starting runtime...
Mar 19, 2015 13:03:53 PM Information [secure-startStop-1] - CORBA Configuration not enabled
Mar 19, 2015 13:03:53 PM Information [secure-startStop-1] - Starting cron...
Mar 19, 2015 13:03:53 PM Information [secure-startStop-1] - Starting registry...
Mar 19, 2015 13:03:53 PM Information [secure-startStop-1] - Starting client...
Mar 19, 2015 13:03:54 PM Information [secure-startStop-1] - The metrics service is disabled for the J2EE edition
Mar 19, 2015 13:03:54 PM Information [secure-startStop-1] - Starting xmlrpc...
Mar 19, 2015 13:03:55 PM Information [secure-startStop-1] - Starting jaxrs...
Mar 19, 2015 13:03:55 PM Information [secure-startStop-1] - Starting graphing...
Mar 19, 2015 13:03:55 PM Information [secure-startStop-1] - Starting solr...
Mar 19, 2015 13:03:55 PM Information [secure-startStop-1] - Starting archive...
Mar 19, 2015 13:03:55 PM Information [secure-startStop-1] - Starting document...
Mar 19, 2015 13:03:56 PM Information [secure-startStop-1] - Starting eventgateway...
Mar 19, 2015 13:03:56 PM Information [secure-startStop-1] - Event Gateway Disabled.
Mar 19, 2015 13:03:56 PM Information [secure-startStop-1] - Starting FlexAssembler...
Mar 19, 2015 13:03:56 PM Information [secure-startStop-1] - Starting .NET...
Mar 19, 2015 13:03:56 PM Information [secure-startStop-1] - Starting Monitoring...
Mar 19, 2015 13:03:57 PM Information [secure-startStop-1] - Starting Monitoring Server on port 5500.
Mar 19, 2015 13:03:58 PM Information [secure-startStop-1] - Starting WebSocket...
Mar 19, 2015 13:04:00 PM Information [secure-startStop-1] - WebSocket server listens on port: 8577
Mar 19, 2015 13:04:01 PM Information [secure-startStop-1] - ColdFusion started
Mar 19, 2015 13:04:01 PM Information [secure-startStop-1] - ColdFusion: application services are now available
Mar 19, 2015 1:04:02 PM org.apache.catalina.core.ApplicationContext log
INFO: CFMxmlServlet: Macromedia Flex Build: 87315.134646
03/19 13:04:02 INFO Macromedia Flex Build: 87315.134646
Mar 19, 2015 1:04:16 PM org.apache.coyote.AbstractProtocol start
INFO: Starting ProtocolHandler ["ajp-bio-8101"]
Mar 19, 2015 1:04:16 PM org.apache.catalina.startup.Catalina start
INFO: Server startup in 67589 ms