Amarië did not follow her love, Finrod, in his journey to Middle-earth. She and Finrod were betrothed, but Finrod never married while in Middle-earth. It is noted in the Lay of Leithian that Finrod was soon allowed to return to life in Valinor and "now dwells with Amarië", so they probably were wed later - but there is no such information in The Silmarillion.
The meaning of the name Amarië is unclear. It is possible her name came from the Quenyan word már, meaning "home", perhaps referring to her home in Aman. It is also possible that her name originated from the Quenyan words mára, meaning "good", and ien, which is a feminine ending.