The Elendilmir was a symbol of royalty among the Kings of Arnor and their descendants; it was a "white star of elvish crystal upon a fillet of mithril".
The original Elendil had descended from Silmariën to Isildur. It was then lost with the death of Isildur at the Disaster of the Gladden Fields and years later it was somehow acquired by Saruman who kept it in Orthanc.
When King Elessar ordered the restoration of Orthanc the original Elendilmir was found. As Elessar took up the kingship of Arnor he wore the original Elendilmir upon his brow. He would only wear it again on high days, whilst on other occasions he would wear the second Elendilmir that had descended to him.