Conquest mode is an interesting feature. It seems as though its effectively a way for Nintendo to gather data on how characters hold up in matches over a period of time.
The way it works is, the game picks specific characters/teams for a period of time, and if you pick any of those characters in a With Anyone match online, you will automatically participate in the Conquest, and your wins as that character will be recorded as part of the Conquest rankings.
Well be running Conquest in beta for a while, Sakurai said on Miiverse. Also, Conquest data where players seem to be losing on purpose will be removed from the calculations. And lastly, theres no point in working on this mode too much on your own, so dont overdo it
Read more at
http://www.siliconera.com/2014/09/16/super-smash-bros-conquest-mode-will-run-beta