John 21

What is the significance of the three different types of love mentioned in these verses? Jesus used a different love in his question to Peter than what Peter answered with.

John 21.15-22

15 ¶ So when they had dined, Jesus saith to Simon Peter, Simon, son of Jonas, lovest (αγαπας, agapas) thou me more than these? He saith to him, Yea, Lord; thou knowest that I love (φιλω, filo) thee. He saith to him, Feed my lambs.
16 He saith to him again the second time, Simon, son of Jonas, lovest (αγαπας, agapas) thou me? He saith to him, Yea, Lord; thou knowest that I love (φιλω, filo) thee. He saith to him, Feed my sheep.
17 He saith to him the third time, Simon, son of Jonas, lovest (φιλεις, fileis) thou me? Peter was grieved because he said to him the third time, Lovest (φιλεις, fileis) thou me? And he said to him, Lord, thou knowest all things; thou knowest that I love (φιλω, filo) thee. Jesus saith to him, Feed my sheep.

 

I have always understood this to be how Jesus was redeeming Peter for denying Him three times, but I don't understand why Jesus used two different "loves" and why Peter answered in the third "love".