"Carne roja" es un término que usan los médicos para prohibir lo que uno no debe comer.
"Carne", in it's standard definition, is the meat of four-legged animals like cow, pig, lamb, goat (in some places horse), venison, rabbit, etc. And it also includes farmyard birds (aves de corral) like chicken and turkey, as well as hunted ones like pheasant, quail, duck....
Fish, shellfish or insects are not regularly considered "carne", except fish, when it's included, together with birds, in the category of "carne blanca".
(Although there was a funny advertising about ostrich meat as "la única carne roja sin colesterol".)
In common speech, people frequently mark a difference between "carne" (the one that walks on four legs), "pollo" (the one with feathers) and "pescado" (the one that swims).