When they had finished breakfast, Jesus asked Simon Peter, 'Simon, son of John, do you love me more than these others do?'
'Yes, Master,' was his answer; 'you know that you are dear to me.'
'Then feed my lambs,' replied Jesus.
Again a second time He asked him, 'Simon, son of John, do you love me?'
'Yes, Master,' he said, 'you know that you are dear to me.'
'Then be a shepherd to my sheep,' He said.
A third time Jesus put the question: 'Simon, son of John, am I dear to you?' It grieved Peter that Jesus asked him the third time, 'Am I dear to you?'
'Master,' he replied, 'you know everything, you can see that you are dear to me.'
'Then feed my much-loved sheep,' said Jesus.
Sadly, most common English translations miss the distinctions between the various kinds of love here. It's usually translated as "Do you love me?" followed by "Yes, I love you."
The first two times, Jesus is asking whether Peter loves with agape, the love of charity and giving and service. For both of those, Peter responds that he loves with filo, the love of a family member or close friend. Jesus' third question, and Peter's final response, are also both about filo.
Jesus is chasing agape, more than filo. It's what he wants from his disciples, especially what he wants from the shepherd of the sheep. It's what he wants from people today, too. His closest disciples are marked by agape for Jesus and for other people.
But does he reject filo? No. It seems that all Peter will commit to is filo, but Jesus doesn't reject Peter for it. Instead, Jesus meets Peter where Peter is and gives him a vocation. If filo is all that Peter can give, then filo will have to do.
In a way, this shows Jesus has high expectations. He wants us to live in agape, a strong love that acts despite feelings. It's a choice, an act of the will. We can do the same thing with filo, but filo is fickle. It's affected by how we feel about a person. You could have a great relationship with someone in your family and do great things for them, but if it turns sour your actions will be soured too. Agape surpasses that.
Jesus' disciples should aspire to act out of agape, but remembering that Jesus won't reject filo. If they result in kindness, goodness, faithfulness, etc, then they'll do. But we should always seek the highest way, the more excellent way of agape (1 Cor 12:31).