A Life Link, simply put, is a link between two people in that if one person dies, so will the other, but both deaths have to occur at exactly the same time.
With Noel Vermillion becoming the Eye of the Azure, she also became immortal. Since Yūki Terumi was able to gain a life link with Noel when she accidentally observed him, neither of them can die. Ragna the Bloodedge shares a life link with Nu-13. A life link in top form, according to Terumi, should make the two people stronger as he comments on how Noel was not able to beat Taokaka in her story mode.