Writing unit tests for existing code can be very simple. You pull up the method (or API or class or whatever), look at the implementation, and start writing tests to make sure you hit it all. Read method, write test, read method, write test. Repeat until desired test coverage is achieved.
Yay! We're done!
Well, no, we're not.
We've tested what was implemented, that's true.
But did we test what was meant to happen? Did we test the intent?
This is a much trickier question. If we're testing a method that's entirely internal - a helper method, say - then the intent and the implementation are probably identical. After all, the consumer of the method is the guy who wrote it, and if it doesn't do everything he wants, then he's probably noticed by now! If, however, we're testing something that is consumed externally - an API, say, or a library - then the intent may be quite different from what the programmer understood.
For example, let's say we have an API that provides some information. The programmer may have implemented that as a GET with url parameters for each data point (e.g., http://mycoolapi?id=1&name=bob&text=hi). The implementation may be perfectly correct. But the intent may be quite different. The product manager may have meant for the API to be a POST with a multi-part form upload, since the next data point is going to be a picture that can be uploaded. Implementation - perfect! Intent - oooops.
So when you're testing something, go ahead and look at the implementation. Just don't forget to look at the intent, too!