Although a loose paper sheet is usually too soft to cut, it can be very thin (sometimes as thin as a razor edge), being then able to exert high levels of pressure, enough to cut the skin. Paper cuts are most often caused by paper sheets that are strongly fastened together (such as brand new paper out of a ream), because one single paper sheet might be dislocated from the rest. Thus all the other sheets are holding this dislocated sheet in position, and the very small part held away from the rest can be stiff enough to act as a razor.
Paper cuts can cause surprisingly acute pain since they usually stimulate a large number of skin surface pain receptors (nociceptors) in a very small area of the skin. Because the shallow cut does not bleed very much, the pain receptors are left open to the air, ensuring continued pain. This is exacerbated by irritation caused by the fibers in the paper itself, which may be coated in chemicals such as bleach. Additionally, most paper cuts occur in the fingers, which have a greater concentration of sensory receptors than the rest of the body.
The random orientation of collagen fibers in skin provides the ability to withstand pinpoint forces. However, skin does not have the same strength against shearing forces, and is easily cut. The same principle can be applied to performers that stand on blades.
A drop of 'liquid bandage' or equivalent onto a papercut can seal the skin to stop pain as well as prevent further opening of the wound (as is common on a location such as a finger).