"someone told me that the moment coefficient was a measure of how stable the aircraft is and where the center of lift is located. Is this correct."
NO!
This is precisely WHY it becomes necessary to refer to real world technical data!
The vast lore of urban legends about aviation is always available from the self-appointed expert. Something on the order of Cliffie in "CHEERS"... but there everyone knows he's a blowhard.
Running into the same situation where fact are requested and not knowing the difference between a snow job and a fact creates a lot of confusion.
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THE pitching moment cofficient is a characteristic of the shape of the airfoil. When an airfoil is mounted in a wind tunnel supported at the 25% mac point, the airfoil may or not "pitch" around that axis.
Symmetrical airfoils do not pitch.
Cambered airfoils do. Generally it is, relative to the 25% point, a nose-down pitch.
The camber can be tailored to eliminate the pitch completely or have a nose-up pitch. The Clark-Y has a nose-down pitch.
Airplanes using it require a horizontal tail. Its derivative the Clark YH has a slightly smaller nose down pitch, the difference being in the aft portion of the profile, and make the shape suitable for flying wings (tailless) but more pronounced treatments of the aft part of the airfoil are now available for those interested in tailless flying.
Before the NACA release of their 4-digit series the C-YH was used for a few fighters in the late 30's.
The amount of the coefficient is a measure of the amount of camber the airfoil has. Some shapes have a small coefficient, others have large values. The coefficient's value is one of the parameters investigated when designing the airplane, as it drives what measures might be required to achieve the performance desired.
The mathematical sign of the coefficient signifies the direction of the pitch, with a negative sign indicating nose-down.
As the camber can be tailored to suit the operational enviroment of the airplane, in the real world it frequently is, with some parts of the wing optimized for low-speed flight, other parts for high speed, with the camber adjusted in the initial design phase to not affect the performance outside the region it is intended for.
High values, which indicate a strong nose-down moment for instance can be used with a blend into a shape with low values, for ease of flight control between landing speed, and cruise speed.
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"center of lift" and its buddy "center of pressure" have been discarded as useful terms by NACA since 1939, with the more accurate identification of the pitching moment, static margin and neutral point now the preferred terminology.
For park-bench aero theorizing, they're OK but they are not scientific.