The Red Dragon was a mob planned by Notch for a future update. However, since Notch's departure from Mojang AB in 2014, it is uncertain whether the plans for this mob died when he left the company. Jeb has stated that if they were ever to be implemented, you would probably get them from a dragon egg.
*The content below is purely speculative.*
It is unknown whether Red Dragons were to be hostile, passive, or tamable, but it can be assumed they would either be hostile or tameable mobs. These dragons would not destroy terrain like the Enderdragon, and would be less dangerous. They would likely appear in the Xbox 360 version of Minecraft as well as the original. Red Dragons would probably be quite rare, and if tamable, they would likely be used primarily as a mode of transportation (probably for flying).