I guess this sentence illustrates quite well the different meanings of these two verbs, so frequently mixed up by English learners.
To listen is to make an effort to hear something, like when we keep quiet for a moment to pay attention to a noise and figure out exactly what it is, or when we select our favorite song on our iPod and enjoy its harmony and lyrics.
That’s why we say we listen to music or listen to the radio, because we’re actively trying to catch the sounds that are being generated by the device. That’s also why you teacher tells you “Now we’re going to to a listening exercise…” – that’s because she wants you to pay attention to what she is going to play…
To hear, on the other hand, is often an involuntary action that happens when sound waves reach our inner ear and do their magic in our eardrums (did you know this word, by the way?). Ex: “Can you hear me? The connection is a little shaky!”
It can also have a similar meaning to the verb to listen, but it’s not used so frequently. Ex: “Hear you mother and take a sweater! It’s cold outside!”
Another meaning of hear is to express that somebody else has told you something, like some juicy gossip, or even some good news. Ex: “I hearshe’s having triplets! I don’t envy her a bit…”
And when your mother/wife/husband/teacher asks you to do something umpleasant, like taking out the garbage or doing your homework, you can always say:
Sorry! I can’t hear you!