Step 1: Unit Testing
Divide your software into components (which can be anything from single functions up to whole programs) and unit-test those components thoroughly, especially as relates to the API and behavior that the rest of the application can see. (Don't forget to check for failure modes too, but beware of binding too carefully to the exact nature of a failure; it's often good enough to just test for the presence of the right class of exception rather than its exact message.) Make sure those tests pass; you're honing them in on a specification of what the component should be doing. (Automated test running helps here, as does a CI system.) This is important because of…
Step 2: Integration Testing
Test that the compositions of components that make up the application work (this is integration testing). Ideally you'll only be finding bugs in the specifications of things at this point (hah!) and wherever you identify that a component is wrong despite passing its unit tests, that tells you that there is a bug. Whenever things fail to work together despite being told to do so, you've probably got a bug in your specs from the previous step so you typically resolve these things by adding more detail to your unit tests and fixing the components until they work.
Note that to make good integration, you want to keep this stage so that the integration itself is sufficiently simple that it is in the “Obviously No Bugs” class of programs instead of the larger “No Obvious Bugs” class. An integration framework like Spring or a scripting language can help a lot here (though with the latter you have to guard against creating components on the sly; if you create a component then admit it and make sure it has a proper usage contract and unit tests to ensure that it meets its contract).
Where you can, you can make components by composing others together; these higher-level components need to be unit tested as characterized in Step 1 above. This might sound like extra work – it probably is – but it does have the advantage of meaning that you can use automated tests for larger parts of the program. (Alas, it's harder to do all integration tests with an automated test tool; such things tend to work better doing unit tests where you can mock out all the irrelevant parts.) But this doesn't save you from…
Step 3: Acceptance Testing
This is where the overall application is tested to see if it actually does what is desired. This might be automatable, but usually isn't. This is level where you bring in users to let them see whether things are what they expected, though you might want to use internal testers a bit first. How easy this all is depends on the nature of the application.
Note also that user interfaces tend to spend more time in this step than the others, precisely because what makes for a good UI is difficult to impossible to pin down in algorithms (it relates much more to human psychology after all).
A final note: What I've written here sounds like testing is meant to be a laborious process that takes ages at the end of a project. It isn't! You can often get parts of an application done before others, do an integration of those parts (with mocks for the other bits) and test quite a bit of how acceptable this sub-application really is. Of course, when doing this take care to stop users from believing that everything is done; one way is to have dialog boxes that pop up and say things like “magic to happen here”. Silly but effective. :-)