鸟禽 is a collective / formal term - "avifauna" rather than "bird". For everyday talk about a single bird, use 鸟 (niǎo). 禽 alone covers domesticated fowl in 家禽 (jiāqín "poultry").
memory
鸟 = bird (specific) + 禽 = fowl (broader category, includes anything with feathers). Together they form a redundant compound meaning "birds in general" - a common pattern in Chinese (e.g. 树木 "trees-and-wood" = trees).
