Numeric Shader Inputs are useful when you want to pass a numerical value to one of your shader parameters and be able to change that value in real time and see the results. You add a Numeric Shader Input from the Shader Path toolbar menu using Shader Input -> Numeric. You can change the Numeric Shader Input's name by double-clicking its label and typing a new name.
You can change the minimum and maximum values at any time and those changes will change the slider's values (and the value being passed to your shader parameter if applicable) immediately. You can also choose whether this control will pass your value as an integer or float depending on what's appropriate for the parameter(s).
To connect this control's shader input to your shader parameters:
When the connection is complete you will see your new connection on the Shader Path and the value will be immediately passed to that parameter every frame (no need to recompile the shader). You can use the slider to see how changing that parameter in real time affects your render.