Alabaster was very expensive in Bible times, and if a person bought an alabaster box, it was only used for very important purposes. So the Pharisees considered it a waste when a woman poured perfume from an alabaster box on Jesus' head in Matthew 26:7-9.
We are assuming here that you are talking about this moment which is also described in more detail in Luke Chapter Six? Here, a woman enters a house owned by a Pharisee and, wordlessly and with many tears, washes Christ with ointment, using her hair to clean his feet. When the Pharisee expresses disgust that such a woman, who, it is inferred, is a prostitute, or certainly a sinner, should be allowed to do such a thing, Christ rebukes him and points out that since the woman has entered, he has done nothing but minister to him while the Pharisee, whose name is Simon, has done nothing. He then turns to the woman and pardons her of all her sins for her great devotion. Since Pope Gregory the Great, this woman has been conflated with Mary Magdelene and so she is often portrayed as carrying an alabaster jar. In fact she is referred to as Mary of Bethany in another Gospel. But then Gregory also conflated Mary of Bethany with Mary Magdelene...
Alabaster historically was a symbol of purity and great honour. It is also thought to have been associated with the Egyptian Goddess Bast (hence alaBASTer). So in some sense, the Woman with the Alabaster Jar may have been associated in the author's mind as connected to ancient forms of worship, particularly of the Goddess. It is significant in this connection that, just before this passage in Luke, the text says 'Wisdom is justified in all her children". Now, the author of Luke would have known what he was referring to as Wisdom ie the Sophia, the Feminine Aspect of God and Bride of Christ. The New Testament, contrary to what most organised Churches will tell you, is full of veiled references to other spiritual cultures - eg Hellenic Alexandrian ones, Egyptian Mysteries ('out of Egypt have I brought my son'), Eleusinian Mysteries (corn and wheat imagery), Zoroastrianism (the Three Wise Men)... So this moment with the Alabaster Jar is very significant in terms of Christ and the Feminine/Sophia. What we are seeing is how, in fact, the Woman with the Jar is much closer in spirit to Christ...
Further, the Alabaster Jar is linked to the image of the Grail. It is the receiving vessel, into which the Holy Spirit pours and the individual can be transformed. Here it is filled with ointment. The woman is not just washing Christ but ANOINTING him. And remember that the word Christ means 'Anointed One'.
So the Jar and the woman who carries it are VERY significant in terms of understanding the inner meaning of the Christ Mystery.