In the anime, Fairy Tail, I know Natsu is well known all around as the Fairy Tail guild's Salamander. However, I've also heard Natsu and Happy refer to Igneel as "The Salamander", specifically in the first episode. Is this a nickname for both of them?
Though IG_42's answer is essentially correct, it is complete.
From memory the conversation went something like this. Very simplistically. [Please edit if somebody has the subs handy :)]
Lucy: I am a mage, looking to join a guild
Natsu: I am looking for Salamander, Though he turned out to be human I am looking for a dragon
Lucy: (All surprised)
Happy: Salamanders are fire lizard so we thought it might be Igneel
Natsu and Happy are very simple minded. It never occured to Natsu that someone else might be using his moniker to fake people. They heard rumours of Salamander and thought it might relate to Igneel.
Real world salamanders have been associated with fire in mythology for centuries. Their tendency to live in old logs sometimes used for firewood led to the belief that salamanders were immune to fire or even born from fire.
Leonardo Da Vinci notably wrote that salamanders only fed on fire.