OP: Hardware calibration generally refers to a device sometimes called a spider (Spyder). You attach it to your computer and monitor in various places and run a program that displays and reads colors on your screen. It generates a profile that your graphics card uses to display optimal screen colors.
AFB: Correct me here if I'm reading this wrong. I thought hardware calibration was calibrating with a hardware device, not something built into the monitor like you describe. Software calibration is using the OS to control the graphics card.
Dale
Both hardware calibration and software calibration use a colorimeter to measure a series of tones and colours to a pre determined standard. This industry standard is called an ICC profile.
Spyder is just one brand. I1 display Pro is generally considered to be the best by most experts. Another is called the ColorMunki display.
With software calibration the colours are controlled by your graphics card.
A more accurate profile is displayed when you have a look up table within your monitor to control your colours, brightness etc. You still use the same device, but rather than the software that comes with the device, you use software that comes with your monitor to acsess the LUT. Colour Navigator 6 or Spectraview would be two of the examples.
This is hardware calibration. Sometimes the colorimeter can be built in to your display like a high end Eizo like the CG277, but that's way out of the OP's budget.