What medium should be kept beneath the paper for papercutting?
Self healing cutting mat



I use this for all my craft cutting. You want to be sure that the surface you are cutting on is completely uniform so as to not cause your cutting hand to hesitate or jump. These mats are for just that. 

Also has the added feature of not cutting up your parent's table. They are usually textured on both side to prevent both the mat and the work from moving around. This also allows the mat to be flipped over and use the other side thus extending the life of the mat. It will eventually wear out though. 



You can usually find these in a variety of sizes to suit your need. Some companies make ones small enough that you can keep in a book if you travel. 

There appears to be, or at least used to be, a small mystery about what these mats are actually made of. One manufacturer describes it as:


  unique composite PVC vinyl material