Why do airplanes use AC electric power?
AC is easier to produce with the engines, that act as generators. 
The engines have a rotating shaft that it is easily equipped with magnetic dipoles all around.



Then, depending on the instrument, the current is either used directly or, by the use of converters, in DC form as not only it is easier to produce, it is also easier to convert. So electronic devices that uses 12, 5 or 3V may contain their own voltage converter (though this is less and less true with modern efficient switch-mode converters). If DC conversion was as much efficient as AC conversion, we wouldn't see high-voltage lines. Last but not least, it's easier to switch, because the current is null twice per cycle. DC must be switched at full current, which is expensive and weight costly. (thanks mins)