The Awestone is an item used to increase in covenant rank in Dark Souls II.

In-Game Description

A stone that is terribly heavy for its size. Offer it to the Victor's Stone to strengthen the bond with your covenant.
The Victor's Stone thirsts for the souls of fallen warriors, condensed within awestones, as if they were once part of the whole.

This stone is incredibly rare to get, and is only obtainable while a part of the Company of Champions.