What is the measurement system used in the aviation industry?
Outside US, SI is generally used for everything except altitude, distance and speed.


Altitude: 1000 ft happens to be reasonable vertical separation, which is somewhat easier to calculate with than the corresponding metric figure 300 m.
Distance: There are two quantities that were never converted to decimal. Time and angle. Since we didn't switch to gradians, it makes sense to keep nautical mile, which corresponds to 1 minute along meridian or equator, as it makes calculating distances from navigational maps somewhat easier. If the maps were marked in gradian coordinates, 1 gradian would be 100 km and kilometres would be used.
Speed: Obviously based on the unit of distance in use.


In Russia, Commonwealth of Independent States (most former Soviet countries) and China (and I am not sure whether some other Asian countries), metric system is used for everything. I believe in Russia they recently switched to flight levels based on feet, but they do use metres below transition altitude.

Before second world war, aircraft built in continental Europe usually had instruments in metric units as well. However before the war aviation was more advanced in the USA, instrument procedures were developed in USA and after the war there was surplus of US-built planes, so most of the world just adopted the US procedures including the units they used as everybody using the same units was more important than any personal preferences. Without these reasons we'd be probably using metric in Europe for everything too.