Aircraft engines are equipped with mixture controls mainly for three reasons, engine optimisation, fuel saving and engine cooling.
Optimum (best power) mixture is used in normal situations and will give the best engine life. Fuel saving and cooling are for responding in emergency situations.
As the aircraft climbs into thinner air, the mixture can be leaned to maintain the optimum mixture.
A lean mixture can give you a little more range if you are about to run out of fuel.
However a too lean mixture can cause engine damage in the long term.
a rich mixture can cool an engine that is close to overheating, but too rich a mixture can cause the spark plugs to foul and cause engine problems as well.
On flights below 5000ft, leaning is usually not required, but can be used, if wisely, to improve engine performance.
At higher altitudes it may even be necessary to keep the engine running.
The best indicators for a correct mixture are CHT, Cylinder head temperature, which is most important but slow to react, and EGT, exhaust temperature, which reacts almost instantly.
Firstly: if the POH has a leaning procedure specific for that engine, that’s what you’ll use.
If not, but the aircraft is equipped with an EGT indicator, lean the mixture until you are at peak (hottest) EGT, and then slightly enrich to obtain the optimum fuel-air ratio and be on the safe side.
If you don’t have an EGT indicator, lean until the engine runs rough, then enrich slightly until it runs smooth again.
If the propeller has a fixed pitch (not adjustable) you can lean to obtain the highest RPM.
In all cases, keep an eye on the CHT temperature after leaning. When in doubt, stay rich of peak.
Enjoy your flights!