What is a Dragonslayer in Fairy Tail?

Answer 1

Dragon Slayers are the people who know Dragon Slaying magic. There are different kinds.

First Generation (Natsu, Gajeel, Wendy) were raised by dragons. Dragons taught humanity the secrets of dragon-slaying magic due to a draconic civil war, whether humans should be treated as prey/slaves or allies. The dragons may have continued to teach humans the magic before the dragons disappeared in year 777, due to a future dragon related threat that they could not deal with (and indeed, one such event occurred at the end of the Grand Magical Games).

Second Generation (Laxus, Cobra) were not raised by dragons, but were implanted with magic crystals called lacrima, these ones contained dragon magic. They share many traits and powers with the First Generation.

Third Generation (Sting, Rogue) were not only raised by a dragon, but also (as they claim) slew their dragon parent to implant dragon lacrima, combining the methods from the first two generations. They've been shown to be able to enter their 'Dragon Force' forms easily and voluntarily. Natsu by comparison had to eat potentially harmful substances to attain that state, namely Aetherion, and Jellal's Flame of Rebuke.

Lastly, the great black dragon Acnologia, was revealed to have been a human king, a dragon slayer who bathed in the blood of so many dragons, that he become one himself, possibly through the power of the legendary dark mage Zeref.