Some do, some don't. Typically the more selective universities will have a supplement (ranging from 1-3 prompts) that is intended to test the applicant's ability to write with its most powerful and economic verse. These supplements, moreover, can set an applicant apart from others as long as the applicant does his/her research and puts time into it.
Now, some of the less-selective schools (state schools typically) won't require them.
In terms of the type of prompt you'll see, it will depend on the university. UChicago is known for creative questions that test a wide array of skills while others simply ask what specific component of said university make it a great place for the applicant to attend. These require research on the university, teachers, classes, curriculum structure, etc. Length should be about a page to ensure an economic usage of language and should be formatted logically.
They come out sometime during the summer and you can check with the specific university with regard to when they do and what is actually required.