to divide a whole into parts: to deal them out as shares, or to be left split against itself
At heart this verb is about splitting a whole into parts. What happens to those parts is what gives the word its range. When the parts are handed out, it means to apportion or distribute: Abraham gives away a tenth of the spoils, Jesus divides the loaves among the crowd, God assigns each believer a measure of faith and a field of work. When the parts instead stand apart, the same word turns negative: a kingdom, a house, or a person broken into pieces that no longer hold together.
The famous saying that a house divided against itself cannot stand turns on this darker use. Anything broken into warring parts collapses under its own weight, which is exactly the point Jesus makes against the charge that he drives out demons by Satan's power: a ruler at war with himself is finished. Paul presses the same word on the quarreling Corinthians when he asks whether Christ has been carved up among their rival factions.
Two near-relatives sit close by. The strengthened form διαμερίζω divides something up and parcels it out among several people, and διαιρέω is the ordinary word for distributing or allotting. Where the stress falls on a violent tear rather than a share-out, the New Testament reaches instead for σχίζω.