I would note all study at the OU is classified as part-time for fee purposes (even if you're studying full time equivalent intensity i.e. 120 credits in a year). You don't apply to the OU as such, you just register for the module you want to take - however some have prerequisites, and if you haven't studied those with the OU they may require you to speak with them and liaise with the department to assess your background and suitability for those modules. This is typically only the case with level 2 modules. Generally the course will be self directed, and they'll send you the materials to study in your own time.
You will have tutorials, which are strictly speaking optional, that you can sign up for throughout the year at scheduled times (they are available both online and in person), and an assigned tutor you can contact for help/advice throughout the course (usually by email but some might also take phone calls). There are also usually module-specific forums where you can discuss topics you are working on with other people taking the module and the various tutors for the module.
For assessment, throughout the year you'll have assessments usually along with a final exam (or for some module an end of module assessment, like a submitted piece of work). Assessments can be computer marked (basically computer based quizzes) or tutor marked (submitted work). Modules seem to often be marked on the basis that your final mark is the lowest out of either your exam or your continuing assessment marks, although sometimes it can be averaged across both (with different weightings for different components).