Paper Stack is an item added by Tinkers' Construct. It is used to make paper tool parts.
Paper tool parts can be used to craft weapons and tools. Although paper tool parts are generally weak, they add another slot for modifiers. Modifiers are items that can be used to boost a Tinkers' Construct tool or weapon power, as well as provide alternatives for enchantments (despite the fact that most Tinkers' Construct items can't be usefully enchanted in the first place).
Crafting a Paper Stack requires 4 paper in a crafting Table