Java web applications are usually packaged to WAR files. WAR is a shortname for Web Application ARchive and physically it's a ZIP archive.
I don't use default manager application at all, because it does not contain much useful information. There's one Java Web application that can do everything manager does and much more. It's called Psi probe and it's a fork of Lambda probe if you ever heard about it.
If you download it, you will download a WAR file. As I said inside is whole web application. Now where to put it? You can either use manager to deploy this WAR file to server, or you can simply copy this file into webapps directory. If you Tomcat is running, it will unpack this file and deploy it on this server. To undeploy it all you have to do is to delete this WAR file and Tomcat will trigger undeploy process.
How do you access this application? Simply goto http://localhost:8080/probe and you will see this web application. You can login using same credentials you used in manager application. Inside you can see deployed applications and much, much more. One of the sweet stuff is inside System Information -> Memory utilization, where you can see how much memory your tomcat currently uses.
Now you see that all directories inside webapps are web applications. Inside docs is current Tomcat documentation, inside examples are some examples, but don't bother with them, most of them are outdated. Inside manager is Manager application and finally host-manager is not very much useful. You can also delete them, this way you will undeploy them.
One application is named ROOT. This application will be presented to the user if he does not go to any particular application.