In Luke 8:2, we meet “Mary, called Magdalene.” St. Luke describes her as a woman “from whom seven demons had gone out.” This tells us two things about her background. She is Mary from Magdala on the shores of the Sea of Galilee (whether by birth or residence, we do not know), and a woman whose wayward life led her to be possessed by seven demons.
Since this identification immediately follows Luke’s account of the sinful woman who anoints the Lord’s feet with expensive ointment, and whom the Lord asks to save some for His burial (Lk. 7:36-50), the sinful woman is often understood as Magdelene.
Mary the sister of Martha and Lazarus of Bethany is likewise often identified as the same person. This Mary similarly anoints the Lord’s feet with a costly ointment, as the sinful woman did and as Magdalene would do of Christ’s Body in death. In this view, Bethany is likely her origin, Magdala where she lived.
In general, the opinions on who Mary Magdalene was tends to be divided along Greek and Latin lines. The Latin or Western Fathers of the Church generally hold that St. Mary Magdalene, the sinful woman, and Mary of Bethany are the same, while the Greek or Eastern Fathers that they are three different women. Scripture provides no clear solution to this question.