I'm a part 61 independent instructor and I keep a 1/2" binder on each student. Why? Because I'm a professional, people pay me fairly well and they line up to fly with me.
1. I have a lesson sheet with the maneuvers we did, and a block checked if it was introduced, a continuation, or completed to standards. This is a reminder for me- the log book has room for maneuvers, but not enough room for comments on them. It's not enough to just write down that pilot has been trained in a maneuver, they need to show ability to complete it to standards. In addition I email students a more thorough debrief with comments on what needs to be improved, what went well, and additional home work. This goes in the same section behind the maneuver sheet and is as much for me as it is for the student. Next time I fly with them, I review this information so I know what needs to be worked on, what we need to discuss prior to flying, etc.
2. Another section has quizzes, such as the pre-solo written, stage exams that I assign, or quizzes from other sources (such as AOPA ASF quizzes). If you don't have copies of these, they might as well have never been done in the eyes of the FAA and the lawyers.
3. Another section for completion certificates for on-line courses I assign, such as AOPA ASF courses.
4. A personal data section. Contains contact information, a copy of pilot certificates, medical, proof of citizenship if required.
Again, I mainly do this for the students. There is a record of what I've done with them so I can review it prior to flying with them. One of the most unprofessional things an instructor can do is to show up not knowing what to do with a student, or not remembering what needs to be worked on or improved. Or, if the student needs to fly with another instructor I can hand the other instructor the folder and they know exactly what I've done. If the student moves or changes instructors, they can have a copy of everything in the folder.
But I also do this because I've sat in on NTSB hearings for friends who were killed. One thing I learned from this is that if a student wrecks an airplane, I will be called in and will be asked detailed questions on the training given. A lack of records is a good way to get noticed by lawyers... if there are no records, there is no proof you did something. And yes, I've had an experienced aviation lawyer (a member of our IAC chapter), look at the records and he concurs that detailed records makes me a more difficult target in the eyes of lawyers.