This achievement is unlocked by defeating 1000 Dragons. You can also check your progress by bringing up the menu, tab over to Quests, access the journal menu, tab once more to achievements, and scroll down to Slayer. This achievement will take some time to complete, but there are plenty of opportunities to kill dragons in Neverwinter. The fastest ways to get this achievement is to slay dragons in the Siege of Neverwinter event. This event normally lasts two weeks and you can get well over 200+ dragon kills here while racking up stronghold resources. The next best event is killing all the dragons in the Well of Dragons. Zone in and see which instance players are gathering in for the run. A run consists of running in a horseshoe shape around the map killing five dragons (Green, Blue, Red, White, & Black). If there are enough people the group may jump to a second instance to repeat the onslaught taking down up to 10 dragons per run. These runs are also helpful in gathering horde treasure for Linu's Favors of which you will need eighty as well as gathering refinement material. Lastly you can just run dragon dungeons and the tyranny of dragons dailies knocking off a few dragons in this manner. All these methods will help get you to 1000. Additionally these are all the named Dragons you can kill towards this achievement outside of the queuing dungeon:
Katatheo the Proud - Stronghold Dragonflight
Oxuno The Treacherous - Stronghold Dragonflight
Chrysos The Vain - Stronghold Dragonflight
Ekdos The Cruel - Stronghold Dragonflight
The Green Dragon Volzakairs - Well of Dragons
The Blue Dragon Kradazuur - Well of Dragons
The Red Dragon Hohengroth - Well of Dragons
The White Dragon Glycamerax - Well of Dragons
The Black Dragon Rogrotus - Well of Dragons
Tiamat the five headed Dragon Queen - Well of Dragons
Halectrizar The Storm Eater - The Siege of Neverwinter Event
Kazlotar The Mordacious - The Siege of Neverwinter Event
Ikzinfraxis The Glacial - The Siege of Neverwinter Event
Phlagenthx The Cremator - The Siege of Neverwinter Event
Venitaxis The Pestilent - The Siege of Neverwinter Event
Zavarak The Thunderous - The Siege of Neverwinter Event
Eziderax The Caustic - The Siege of Neverwinter Event
Glaxenheim The Pale - The Siege of Neverwinter Event
Akregon The Pyre - The Siege of Neverwinter Event
Toxicallus The Despoiler - The Siege of Neverwinter Event
The Red Dragon Karrundax - Caverns of Karrundax (Level 54, 3 Player Dungeon)
The Red Dragon Lostmauth - Lair of Lostmauth (Level 70 Epic, 5 Player Dungeon)
The Blue Dragon Fulminorax - Malabog Castle (Level 70 Epic, 5 Player Dungeon)