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It means various things to various people.
However, in principle is a way of applying a slowly-varying voltage to a circuit in order to reduce the starting current, overshoot, etc.
I assume you are referring to a motor. When you first power it up, it is not moving, therefore the back EMF is zero and hence the entire supply voltage is applied simply across the DC resistance of the motor (armature). Since that resistance is low, the curent is very high, until the motor accelerates and the back EMF is significant.
Look at the equation;
Va is the armature voltage, Ra, La are the resistance and inductance of the motor, Eo is the back EMF. Clearly, when Eo is zero (when the motor is stalled or not yet spinning) the armature current Ia is very large (the armature time constant is much smaller than the time it takes the motor to accelerate and so the current rises rapidly to the value Ia=Va/Ra)
The solution is then to apply an armature voltage Va that starts off from a low value, such that the current is maintained within reasonable limits. Then, as the motor accelerates and the EMF Eo builds up, the Va voltage is slowly increased, but maintaining the Ia current within a safe limit.
This is then soft-start: applyt he voltage slowly, as opposed to a hard-start, where the full voltage is applied full-blast.