Yes, crows (and other birds in the crow family such as jays and ravens) are a passerine, which means that they are a bird in the order Passeriformes. You can see a very nice taxonomic breakdown at Animal Diversity Web.
People are often surprised that birds in the crow family (corvids) are songbirds, oscines: birds with a complex voice box that can learn their songs. Crows often make "squawky" kinds of noises rather than singing elaborate songs, however this doesn't mean they aren't capable of learning and making a whole variety of sounds. Many corvids are actually vocal mimics! I've heard Eurasian jays mimic the wind, the radio, and jackhammers (yep there was lots of construction around them at the time). And they continue to learn new vocalizations into adulthood.