Cornell's free bird ID app, Merlin, has a new "Sound ID" feature. It helps you identify nearby singing birds.
Be mindful that playing bird songs outdoors stresses birds in breeding season. A bird will perceive another bird's song as another male threatening its territory. See the American Birding Association's Ethics page below.
Birdsong for the Curious Naturalist by 
The Singing Life of Birds by Not all nature sounds are birds! Eastern Chipmunks, American Toads, Wood Frogs and other animals also vocalize and can sound like birds.