Thanks to adarshr, I was able to look at the Problems window and determine that the build was failing because it could not find a class I had written.
I had used the MS TFS plugin to create a "shelveset" and it was supposed to have removed my pending changes in the process. However, this integration with the TFS snapin and Eclipse is obviously not well implemented, since the Eclipse project still thought the file existed and was complaining that it could not be compiled.
I went and manually deleted those "files" or "non-existing files" from the Eclipse project (that I thought I had removed with the shelveset action) and the problem was solved.
Also....
Another annoying things is that the Tomcat error I was getting by trying to debug within Eclipse was like this:
SEVERE: Error configuring application listener of class
com.CompanyName.ProjectName.servlet.StartupConfigListener
java.lang.ClassNotFoundException:
com.CompanyName.ProjectName.servlet.StartupConfigListener at
org.apache.catalina.loader.WebappClassLoader.loadClass(WebappClassLoader.java:1643)
In reality, there was no problem at all with StartupConfigListener.java!! The build failed due to the OTHER problems I mentioned above, and therefore I guess this was the first class it attempted to load and failed.... because the entire project hadn't been able to compile perhaps?
(Here's to hoping my next project is using Visual Studio instead of Eclipse!)