We are taught to lean the engine during the cruise. We are also taught that if we over-lean, the engine can overheat. The engine is specifically prone to overheating during the climb because of the high power and low airspeed (which is why we always select the mixture to full rich before climbing).
What we are actually talking about when we use the word "overheat" in this sense is the cylinder head temperature being too high. Therefore, it follows that if we have a CHT guage, we can use it to lean the engine far more accurately - which probably means leaning more aggressively (since we can be more certain that we haven't over-leaned) and therefore saving fuel.
Of course, each cylinder will be at a slightly different temperature because of the way the cooling air flows around inside the cowling. Some designers will decide which cylinder they think should be the hottest, and put the guage on that cylinder. Those designers who think their customers have lots of money will put a guage onto each cylinder so that the pilot can choose to look at the hottest cylinder himself when leaning.
That's a basic overview, but since no one has ever trusted me with an aircraft with such a guage, and I can't afford to pay to fly one myself, I'll leave the details for someone who understands them! Just one thing to add, though, which is to not confuse the CHT with the EGT. Both of them can be used to lean the engine, but the technique is different. And EGT guages are much cheaper (but, in my experience, rarely work).
FFF
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