The Kabbalistic Wedding Ring
According to the Zohar (Tikkunei Zohar, Tikkun 5), the wedding ring should be square on the outside as the Hebrew letter 'Final Mem' ם. One of the explanations for this is that the four corners represent the Tetragrammaton, YHVH יהו"ה, the holiest name of God. This is also a symbol for the vessel for bliss that is going to be fulfilled after the wedding.
The inner circle represents equality and unity. After the union, the bride and the groom should see each other as one, the same as in the circle there is no difference between up and down, right and left.