The Minute Paper is an effective tool that has a variety of potential pedagogical aims. It has been an important part of my approach.
The way I've used it is as an assignment due no later than 24 hours after each class meeting. Then, as TA I review the responses before the next class meeting as I finalize the lesson plan. My review process involves giving a sentence or two of feedback to each minute paper and making a list of 10 key themes. Here's how I typically use Minute Papers:
I open class with a "micro-lecture" or simply a recitation of the major themes and questions of the Minute Papers. This sets the tone for the class that the classroom space as a productive theoretical space is co-created. It gets the wheels turning and refreshes in everyone's mind the topic at hand. Often, if I recite the Minute Paper themes at the beginning of class, each theme gets a descriptive sentence (i.e., what the Papers say) and a perscriptive sentence (i.e., how this connects to a key course theme, or, to today's lesson.)
Even if you don't use Minute Papers like this, it's an effective tool to gauge student comprehension of course material, the effectiveness of pedagogical tools, and keep students engaged in wrestling with course material after class time.
I typically use the Minute Paper structure as presented by Tufts University's Center for the Enhancement of Learning and Teaching (CELT), adapted from Angelo and Cross (1993). I was influenced by Professor Gregg Lambert's use of this method in his mixed undergraduate-graduate philosophy seminars.