Familiarity is an Additional Stat in Nioh. It indicates how much you have used an item. The higher your familiarity, the more Amrita you will obtain when you offer the item at a Shrine. Every item has level at which their familiarity is maxed out.
The concept of Familiarity is inspired by the traditional Japanese belief that every day objects have a spirit residing in them. So the longer you use a weapon, the stronger it gets, with that spirit making a deeper connection with the user. Therefore, the decision to part with it will be a difficult one, but one will be rewarded in kind.
Familiarity is a replacement for weapon durability which was present in early development and in the alpha demo of the game. Weapons would degrade in durability over use, diminishing their effectiveness. The weapon could be repaired or discarded for a new one.