Stiffen, tighten, and harden are all terms for the same thing. You may also hear it called "increasing the rate".
This means to make the springs harder to compress. This limits how far the wheel travels up and down for a given bump, and how big a bump must be hit before the wheel moves. Very stiff springs will make the car corner flatter and respond to the steering, gas, and brakes more quickly. However, they will also make the car unsettled over rough pavement, curbs, etc.
Softening, loosening, or reducing the spring rate just means the opposite. The car will absorb more bumps without getting unsettled, but it will not be very responsive to control inputs.