Hi there, The bird-type creatures that the Na'vi fly on are called Lavi.
Actually the Na'vi tame creatures they call Ikran. Humans call them mountain banshees. The question was regarding the "big red bird" likely refers to the great leonopterex or toruk (meaning "last shadow") that Jake temporarialy bonds with to fight the RDA.
The 'red bird' refered to is most probably the Great Leonopteryx. It's latinized name as given by the scientists on Pandora is Leonopteryx rex which means something along the lines of 'flying king lion'. The Na'vi name for the creature is Toruk, which means 'last shadow'. The animal is feared and admired by the Na'vi, forming an integral part of their mythology concerning the Toruk Makto, a 'legendary' rider that would ride this beast. It has only happened six times so far, with Jake being the sixth. [An Activist's Survival Guide by Wilhelm & Mathison pg. 78-79]