翼 is literary and formal. In everyday speech, 翅膀 (chìbǎng) is the common word for wing. 翼 appears in formal or compound words: 机翼 (jīyì, aircraft wing), 左翼 (zuǒyì, left wing), 右翼 (yòuyì, right wing).
grammar
小心翼翼 (xiǎoxīnyìyì, extremely cautious/careful) is a common idiom using the reduplication of 翼. It literally means 'careful with careful wings' — acting with extreme caution.
