You'll be introduced to the vast majority of the game's Elder Dragons during and after the assingment A Wound and a Thirst (although Low Rank Kirin can be hunted earlier), after which the main story assignments will take you through several investigations that culminate in hunts for other Elder Dragons. Of the non-story monsters you can regularly fight, the following are Elder Dragons: Kirin, Kushala Daora, Nergigante, Teostra and Vaal Hazak, and you'll need to fight each of them numerous times to gain materials necessary to craft late-game gear. Kill any fifty Elder Dragons to get this trophy/achievement.
Tempered Elder Dragons are also the best source of the rarest drops in the game - Warrior's Streamstones and Hero's Streamstones. That being the case you'll ideally earn a lot of these kills as you hunt Tempered Elder Dragons, which is a hobby you can indulge in after the assignment Thunderous Rumble in the Highlands.