Dragon Slayer Magic is a form of "Lost Magic" originally taught to humans in order to defeat real dragons.
These mages have the ability to replenish their magic by engulfing their respective element. In rare occasions, they can eat other elements, but this will cause great agony. So far, Natsu and Gajeel are able to utilize other elements. Natsu can use lightning and Etherion magic while Gajeel is able to use shadow magic.
According to Rogue, it is mentioned that all generations of Dragon Slayers have the motion sickness. For Cobra (second generation), it isn't confirmed because he is always riding a snake and not using any other transportation.
These individuals are taught by their dragons.
These individuals have Lacrima installed in them, providing their Dragon Slayer Magic.
These individuals are taught by their dragons and have Lacrimas inside them. In addition, they have killed their dragons.