Try not to use Mochitest
Yes, really. For many things Mochitest is overkill. In general you should always try to use one of the lighterweight testing frameworks. For example, if you only want to test a single XPCOM component then you should use xpcshell. On the other hand there are some things that Mochitest cannot do, or isn't designed to do. For example, for visual output tests you should try to use the reftest framework. For more information on the different types of automated testing frameworks see Mozilla automated testing.
To run the Mochitest tests you need to build Mozilla, change directory to objdir/_tests/testing/mochitest and run the command:
> perl runtests.pl
This will open your new build with a document containing a "Run Tests" link at the top. To run the tests simply click that link and watch the results being generated in front of your eyes.
Writing new Mochitests
A basic document on Writing MochiTest-based unit tests has already been started. (Maybe the basic stuff from that document should be moved here and that document should give the matter more detailed treatment?)
Adding new Mochitests
Once you've written new tests you probably want to make sure they'll be run automatically by the Mozilla tinderboxen. Maybe someone could add something here about where they should be added, and what you need to do to make sure the tinderboxen will be running them?
For answers to frequently asked Mochitest questions see the Mochitest FAQ. (Information between these two documents could really do with a lot of reorganization.)