A picket line is a horizontal rope along which horses are tied at intervals.
The rope can be on the ground, at chest height (above the knees, below the neck) or overhead.
The word picket (meaning sharp point) in this context refers to one of the pins that are used to secure the line.
Picket pins were originally pointed wooden stakes hammered in to the ground.
[1] To secure a single horse, only a single picket pin is needed to which the horse is attached via a length of rope attached to a hobble on one leg.