Update:

Some 10 years later perhaps the best way to test a private method, or any inaccessible member, is via @Jailbreak from the Manifold framework.

@Jailbreak Foo foo = new Foo();
// Direct, *type-safe* access to *all* foo's members
foo.privateMethod(x, y, z);
foo.privateField = value;

This way your code remains type-safe and readable. No design compromises, no overexposing methods and fields for the sake of tests.

If you have somewhat of a legacy Java application, and you're not allowed to change the visibility of your methods, the best way to test private methods is to use reflection.

Internally we're using helpers to get/set private and private static variables as well as invoke private and private static methods. The following patterns will let you do pretty much anything related to the private methods and fields. Of course, you can't change private static final variables through reflection.

Method method = TargetClass.getDeclaredMethod(methodName, argClasses);
method.setAccessible(true);
return method.invoke(targetObject, argObjects);

And for fields:

Field field = TargetClass.getDeclaredField(fieldName);
field.setAccessible(true);
field.set(object, value);

Notes:
1. TargetClass.getDeclaredMethod(methodName, argClasses) lets you look into private methods. The same thing applies for getDeclaredField.
2. The setAccessible(true) is required to play around with privates.