to break a whole into parts that go separate ways: to share something out among several people, or to leave a group split into opposing sides
The picture behind this word is simple: one thing, several parts, and the parts end up apart from each other. It is a stronger relative of a plainer verb for dividing (μερίζω), and the strength shows in where the pieces go, into separate hands or onto opposing sides.
Sometimes the sharing out is a good thing. At the last supper Jesus hands over the cup and tells his followers to divide it among themselves, and the first believers in Jerusalem sell what they own and share the money out to anyone in need. Here the word describes generosity: one thing broken up so that everyone gets a portion.
Sometimes it is a wound. A kingdom split against itself falls, Jesus says, and he warns that his coming will leave five people in one house set three against two, father against son and mother against daughter. Nothing is being shared in those verses. The same act of breaking a whole into pieces has turned destructive, because what is broken is a family or a nation rather than a loaf or a purse.
The crucifixion is where the two shades meet. All four Gospels use this word for the soldiers dividing up Jesus' clothes, echoing the line in Psalm 22 about parting a victim's garments. It is a share-out, but a share-out of plunder taken from a dying man, and John sharpens the point: the soldiers divide the clothes with this word, yet refuse to tear (σχίζω) the seamless tunic and gamble for it whole instead.