Think of a module as being a place to wire up a number of other things, such as directives, services, constants etc. Modules can be injected into other modules giving you a high level of reuse.
When writing an angular app, you would have a top-level module which is your application code (without templates).
Services are mainly a way to communicate between controllers, but you can inject one service into another. Services are often used as a way to get to your data stores and people will wrap the angular APIs, such as ngResource. This technique is useful since it makes testing (particularly mocking) quite easy. You can have services for doing other things like authentication, logging etc.
Directives are used for creating widgets or wrapping existing things like jquery plugins. Wrapping existing plugins can be a challenge and the reason you would do this is to establish a two-way data binding between the plugins and angular. If you don't need two-way data binding then you don't need to wrap them.
A directive is also a place for doing DOM manipulation, catching DOM-events etc. You should not be doing DOM-related stuff in controllers or services. Creating directives can get pretty complex. IMHO, I recommend first looking at API for something that can do what you are looking for OR ask Angular's Google Group for advice.