Turns out my instance of mongod wasn't running at the time I was executing the commands.
Make sure to run "mongod" in your console at all times you're using the server.
The second error I encountered was failing to specify the host and port of the MongoClient, in the following way:
MongoClient mongo = new MongoClient();
Instead you can do:
MongoClient mongo = new MongoClient("localhost", 27017);
You can stick that into your servlet context and pass it around as needed to other servlets, and while I have declared MongoClient mongo with the new statement here, I would recommend keeping it as a class variable of the ContextListener.