The woman is not named in Luke's Gospel, but she is traditionally Mary Magdalen.
37 suddenly a woman came in, who had a bad name in the town. She had heard he was dining with the Pharisee and had brought with her an alabaster jar of ointment.
38 She waited behind him at his feet, weeping, and her tears fell on his feet, and she wiped them away with her hair; then she covered his feet with kisses and anointed them with the ointment.
39 When the Pharisee who had invited him saw this, he said to himself, 'If this man were a prophet, he would know who this woman is and what sort of person it is who is touching him and what a bad name she has.'
40 Then Jesus took him up and said, 'Simon, I have something to say to you.' He replied, 'Say on, Master.'
41 'There was once a creditor who had two men in his debt; one owed him five hundred denarii, the other fifty.
42 They were unable to pay, so he let them both off. Which of them will love him more?'
43 Simon answered, 'The one who was let off more, I suppose.' Jesus said, 'You are right.'
44 Then he turned to the woman and said to Simon, 'You see this woman? I came into your house, and you poured no water over my feet, but she has poured out her tears over my feet and wiped them away with her hair. 45 You gave me no kiss, but she has been covering my feet with kisses ever since I came in.
46 You did not anoint my head with oil, but she has anointed my feet with ointment.
47 For this reason I tell you that her sins, many as they are, have been forgiven her, because she has shown such great love. It is someone who is forgiven little who shows little love.'
48 Then he said to her, 'Your sins are forgiven.'
49 Those who were with him at table began to say to themselves, 'Who is this man, that even forgives sins?'
50 But he said to the woman, 'Your faith has saved you; go in peace.'
Powerful stuff. Thanks so much for pointing out this passage as well. Now including my other post in response, we have two biblical passage referencing this arena.