To get this Achievement, you must defeat 50 Elder Dragons.
Elder Dragons are the end game monsters that you will fight (Kirin, Nergigante, Kushala Daora, Teostra, Vaal Hazak, and Xenojiiva).
You CANNOT capture an Elder Dragon, as such all 50 must be killed.
You DO NOT have to be the one to kill the Elder Dragon, as long as you were in the game, all members will be credited.
All quest versions (Assigned, Optional, Investigation, Expedition, Event) will count towards your 50.
Normally you can only hunt one Elder Dragon at a time but there are a few high HR Investigations that will allow you to fight two at once.
**Currently 4-11-18 the Cherry Blossom Event is in play and one of the Events you can post is a Two Kirin Quest in Low Rank which I highly recommend farming to get your 50 much faster!