In terms of paper,
Paper is made of cellulose, and in papers there are two real main kinds of bonds, there are the strong intermolecular forces, such as hydrogen bonding between the microfibrils of cellulose, and there are weak intermolecular forces such as Van Der Waal’s forces. When you cut with scissors on paper the bonds that are going to be broken are the weak intermolecular forces.
The pressure created by the scissors deforms the material by poisson expansion (which essentially is when you apply pressure to the middle of something, it will split in both perpendicular directions) and breaks the weaker intermolecular forces in the material because the molecules with those bonds begin to become very distorted and out of shape. The weaker bonds will break and create two seperate parts of the material with the odd strong bond in the way being broken too.
The reason you’re not able to reattach the two seperated bits of paper is that the layer of atoms exposed will oxidise very rapidly and for a barrier that stops reintegration. (Fun fact, this is the reason cold welding exists as in space the oxide layer can’t form and you can just put two pieces of metal together) – see this for more details http://www.coolsciencefacts.com/2006/metal.html