"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.