Vocal learning - the ability to imitate complex vocalizations - is a relatively rare ability in the animal kingdom. Humans obviously are excellent vocal learners, and this ability is central to both singing and speech. Surprisingly, vocal learning of complex sounds (like speech or songs) has not been found in ANY other nonhuman primate.
However, vocal learning is common among birds, and at least three major bird groups have evolved vocal learning, probably independently: the songbirds (oscine passerines), parrots, and hummingbirds. This has led to songbirds becoming the major group in which the genetic and neural basis for vocal learning is studied. Unfortunately, however, birds have both a very different brain from that of mammals, and a completely novel vocal production system (called the syrinx). Thus, there may be important differences between vocal learning mechanisms in humans and birds.
Another large group of vocal learners are the cetaceans: whales and dolphins. Again, unfortunately the mechanism dolphins and other toothed whales use to make sounds is evolutionarily novel, and unrelated to the human vocal tract. Cetacean brains are also rather peculiar, with a very thin cerebral cortex. Thus, the similarities between cetacean vocal learning mechanisms and our own may be quite circumscribed.
Are there ANY animals capable of complex vocal learning, that have brains and vocal tracts like ours? Yes: many seals are capable of vocal learning, and they produce vocalizations with a normal mammalian vocal tract and larynx (just like ours) and have a quite ordinary mammalian brain. (The most famous example is Hoover, a harbour seal who could speak). They are also relatively common, small (compared to humpback whales or elephants, another potential vocal learner) and very easily trained to vocalize.
Thus, pinnipeds (seals, sea lions and walruses) look like the best group of nonhuman animals to help scientists learn what's involved in complex vocal learning at the physiological, neurological, and genetic levels. You may be surprised to know that we know very little about vocal production or vocal control in this group. This is something that my colleagues (Colleen Reichmuth at the Pinniped Lab UCSC and Terry Deacon at UC Berkeley) and I are hoping to change with our upcoming research on pinniped vocal control!