For example, in straight-and-level flight, the power's primary instrument is the Air Speed Indicator (ASI), and the supportive are the engine instruments.
In another example, straight constant-airspeed climb, the primary are engine instruments, and the supportive is the ASI.
Why does the primary instrument change?
Because during certain maneuvers, certain instruments provide more relevant information to a pilot than do others.
In straight and level flight, the heading indicator provides more relevant bank information as it tells you whether your current bank attitude is maintaining the desired heading. The altimeter is providing the most relevant pitch information as it’s telling you whether your current pitch will maintain the correct altitude. The airspeed indicator is the best indication for your required power as it’s telling you if the current power settings will maintain the desired airspeed. Other instruments provide supporting information. Here in straight and level flight the attitude indicator provides good supporting bank and pitch indications, the VSI provides supporting pitch information, etc. Supporting instruments change to primary instruments in certain maneuvers and vice versa.
And, incidentally, in a established constant airspeed climb, your primary for bank is the heading indicator and your primary for pitch is the airspeed indicator. Primary for power is either the tachometer, manifold pressure, torque, or fan speed gauges, depending on the aircraft you are flying. Typically you would set to maximum continuous power in a light airplane for this situation.
The whole point of classifying primary or secondary instrument (or the other method control-performance) is to teach pilots to look at what is important during different phases of flight.
When you read the Instrument Flying Handbook, it will tell you what instruments are primary and what instruments are secondary.
For example, in straight and level flight, the altimeter is primary for pitch, the airspeed is primary for power, and the turn and bank indicator is primary for bank. If you examine what these instruments are doing at this exact moment, none of those instruments should be moving... they are constant. Hence, the primary characteristic of primary instruments are they are constant.
Secondary instruments for straight and level flight are the VSI for pitch, power for airspeed and attitude indicator for bank. These instruments show you how to control the airplane to return to the aircraft flight attitude you had before. For example, you notice the airspeed start slowing. You take a look at the secondary instrument, power and notice that it has slowly backed off. Increasing power should then cause the airspeed to return to normal. Hence, the primary characteristic of secondary instruments are they show trends.
Reading the manual, you will also discover that anytime you change the pitch or bank of the airplane you should look at the attitude indicator. It is primary instrument for all changes in the flight attitude of the airplane. For example, if you want to change between straight and level to a constant airspeed climb, you will use the attitude indicator to set the pitch attitude and then verify the airspeed is climbing at the correct speed.
The answers already provided are very good, this answer is meant to say the same thing in different terms, not to contradict what has already been written.
A primary instrument is one that provides a value for you to hold. If you want to maintain 5,000 feet, then the altimeter is the primary instrument because though there are other instruments that can tell you about your altitude, the altimeter is the only one that can show you what your altitude is.
Similarly, when entering a climb, the attitude indicator is primary for pitch, because you should be setting a particular pitch and the attitude indicator is where you look to see what degree of pitch you have.
Primary instruments give you a target, supporting instruments help you hit it.
All the answer except for quiet flyer's are spot on. Although, I do appreciate, acknowledge and understand his point of view. 90 years of instrument flight has dictated which instrument is primary and which is secondary for each state of flight. It is not arbitrary. The way to look at it is that you are trying to achieve a certain outcome without the benefit of your usual feedback. You can not rely on on sight, sound (in some cases), touch (seat of the pants), or sense of motion and balance.
Say for instance, if you are flying IMC. And, you want to fly straight and level in unaccelerated flight. The Airspeed Indicator is your primary for power because (hyperbolically) who cares how fast your prop is turning. Your RPMs are not an outcome. At least, it is not your desired outcome. Your desired outcome is a constant airspeed. Your engine performance is tangential to that. In fact, your RPMs are going to change as your Density Altitude changes even at the same airspeed. Then, throw into the mix a constant speed or otherwise adjustable prop, your Tach becomes almost unusable as a primary for power except when you are changing from one maneuver state to another.
Personally, I think that the Control and Performance Method is much better at addressing all of the posters' views. Especially quiet flyer's. In reality, everyone that I fly with uses Control and Performance intuitively. In VMC, Control and Performance is automatic and second nature because you are using sight, sound, and sense of motion primarily to fly the plane. It can still be used when those senses are not available in IMC.
Where the Primary and Supporting Method shines is when you start losing the use of your instruments in partial panel flight. It tells and teaches you what instruments (supporting instruments) can be used when the ones you want (primary instruments) are not available or inop.
"Why can't we use just one as a primary instrument?" -- sometimes we can. I've seen video of a Cessna 120 being flown in cloud with all gyro instruments covered up. The pilot's hands were not touching the control stick or throttle. The airspeed, tachometer, altimeter, and vertical speed indicator (if present) were not covered up, but they might as well have been. The aircraft was trimmed for a controlled descent in fairly smooth air and emerged from the bottom of cloud layer about 5 minutes after the video started. The magnetic compass was obviously being used as the primary instrument. Steering corrections were accomplished by using the rudder pedals to bank the plane slightly via the "dihedal effect" created primarily by the aircraft's high-wing configuration. Don't try this at home unless you understand what heading to use, and why.
This whole business of classifying instruments into "primary" and "supporting" is rather subjective and arbitrary. Who gets to decide that the "in straight-and-level flight, the power's primary instrument is ASI", rather than, say, the tachometer? Maybe a lower-than-expected airspeed in level flight is caused by ice on the wings, or by the pilot forgetting to raise the landing gear, rather than by poor engine performance.
