An alternating-possession in basketball is a method of putting the ball into play via an inbounds pass rather than a jump ball. It is determined by utilizing a possession arrow and is usually only used in non-professional basketball leagues.
The rule was not created until 1981, and previous tie-ups up until that point would always be settled by a physical jump ball. The rule was first introduced to college basketball, and almost every level below college uses the system.
The main reasons for the rule's implementation were to speed up gameplay and not to reward taller players who could easily win every jump ball situation they got into. Alternating the possession is a more fair way to resolve essentially tied situations in a game.
The first team that gets the benefit of the arrow is whichever team loses the jump ball. This means that the next jump will automatically give them possession and will continue switching back and forth. The team that has the possession arrow at the end of a quarter or half will start with the ball in the next period.
The other situation in which the possession arrow is utilized is in regard to unclear out of bounds plays. If a review is inconclusive, or the basketball game does not have a review system and the referees cannot decide on a call, then the possession will go to whoever has the arrow rather than forcing the refs to make a call they are not sure of.