The spring rate is a measure of how stiff or soft a spring is. In other words, how much a spring resists being compressed at any given moment.
When the spring rate is changed uniformly across all four wheels, the vehicle will behave more "hard" or "soft". If the front and rear are adjusted differently, the balance can be changed and cause more oversteer or understeer, depending on how they are adjusted.
Designing a Spring With Different Spring Rate
From the production side, a spring is made stiffer or softer based on how thick the metal is that is used to create the spring, the type of metal used (almost always spring steel), the diameter if the spring, the length of the spring, and the angle of the coil. Size factors tend to play a role in which of those can be altered and how much.
In addition, progressive rate springs change one or more of those, gradually along the length of the coil.