The repetition of other people's words or sounds is echolalia. When the toddler you babysit repeats everything you say, over and over again, you can call it "annoying," or you can call it echolalia.
Echolalia is a psychiatric term that's used to describe what some people with mental disorders or autism tend to do, automatically repeat what they hear other people say. There's no meaning intended in echolalia — it's simply a mechanical echoing of sounds. Babies do this too, when they're learning to speak. The word echolalia combines the Greek word for "resound, or echo," with lalia, or "speech."