A PID Loop in HVAC is really just a module trying to close the gap between what you want and what you have. It continuously compares the setpoint to the measured value and reacts to the error between them. If there is no error, the Loop has nothing to do, and its output may remain static.
The Proportional (P) term reacts to the size of that error. If the temperature or pressure is a long way from setpoint, the Loop makes a big change. If it is only slightly off, it makes a small change. This gives an immediate response and is what makes the system feel responsive.
The Integral (I) term looks at error over time. If the system has been a little bit wrong for a long time, the integral action slowly builds up and nudges the output until the error disappears. In HVAC, this is essential for removing steady offsets, such as a room that always sits half a degree below setpoint.
The Derivative (D) term looks at how fast the error is changing and tries to react in advance. In theory, this can help dampen overshoot. In practice, HVAC systems are slow and sensors are often noisy, so the derivative term tends to react to noise rather than real system behaviour. For that reason, it is rarely used in building control, and most stable systems operate as simple PI loops.
One of the most important parts of tuning a loop is understanding how quickly the system responds. A space temperature may take many minutes (or hours) to react to a heating change, whereas an AHU supply air sensor may respond in seconds. Applying aggressive tuning to a slow system will cause hunting and overshoot, while gentle tuning on a fast system can make it feel unresponsive. Good control matches the tuning to the natural speed of the plant.
It is also worth remembering that the Loop only sees numbers. It does not understand engineering units, only the size of the error. If you are controlling pressure in bar and the typical error is 0.1 bar, the Loop is trying to work with very small values. In some cases, multiplying the input and setpoint (internally only, before the Loop), by the same factor, for example multiply them by 10, improves resolution and makes control smoother without changing the real behaviour of the system.
At its heart, PID control in HVAC is about respecting the physics of the system, choosing the right level of complexity, and giving the Loop sensible values to work with.