"Their next turn" refers to the next immediate turn the target would be able to take their action.
Don't forget in 5e, a turn is defined as a single person's action. So each round consists of as many turns as their are characters (PC's and DM controlled monsters/characters) in the encounter.
A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other. -Roll 20, Combat
So if a Cleric casts "Command" on a target that has already taken their action/turn during the current round, the spell will take effect during the target's action/turn on the next round. If the Cleric casts "Command" on a target that hasn't taken their action/turn during the current round, the spell will take effect on the same round the Cleric cast the spell, on the target's action/turn.
As far as the spell being cast as a reaction to the target's turn, the target will still be able to act during this turn as planned, but the spell will effect them on the next turn the target takes.