Unit tests are simply a way to exercise a given body of code to ensure that a defined set of conditions leads to the expected set of out comes. As Steven points out, these "exercises" should check across a range of criteria ("BICEP"). Yes, ideally you should test all of your classes and all of the methods in these classes although there is always some room for judgement: testing shouldn't be an end in itself but rather should support the wider project goals.
Ok, so...theory is nice but to really understand Unit Testing, my recommendation would be to pull together the appropriate tools and just get started. Like most things in programming, if you have the right tools, it is easy to learn by doing.
First, pick up a copy of NUnit. It is free, easy to install and easy to work with. If you'd like some documentation, check out Pragmatic Unit Testing in C# with NUnit.
Next, go to http://www.testdriven.net/ and get a copy of TestDriven.net. It installs into Visual Studio 2008 and gives you right-click access to a full range of testing tools including the ability to run NUnit tests against a file, directory or project (typically, tests are written in a separate project). You can also run tests with debugging or, coolest of all, run all the tests against a copy of NCover. NCover will show you exactly what code is being exercised so you can figure out where you need to improve your test coverage. TestDriven.net costs $170 for a professional license but, if you are like me, it will very quickly become an integral tool in your toolbox. Anyway, I've found it to be an excellent professional investment.
Good luck!
