Do I need to write a resume too if I am filling in a job application?

Do you NEED to? NO

However, you should.

Heck, you don't even have to fill in the entire job application to be considered.

While the info might be the same, I think it will look better on you if you attach your resume to the job application.

Hi,
I agree with CaptainForest; you should submit a resume.
There are many sites online with different types of resumes to use.
If the job application isn't going to be faxed anywhere, such as from a State Employment Agency, or Employment Commission, then I have written in the application with words such as "Please see Resume". Then attach the resume with the job application.
This saves a lot of time with applications such as headings of:
Education; instead of writing all this down again, just use the above phrase.
Past Employment.
References, both work and non-work.
I do wish you the best of luck, and the more applications you fill out, the better chances of landing a job.