Japan has a long tradition of audience participation in performance, from Matsuri to Baseball Games. In Idol Culture the fans cheer on the idols during the songs. Its not required to be a fan, but its fun.
There are different types of calls and mixes. This is an ever evolving culture surrounding fancalls and mix chants.
English Mix
Japanese Mix
To begin understanding, please watch these videos:
The man who popularized MIX in Japan explains what MIX is, Encho's Video on MIX (Producer of Jewel Kiss)
A collection of various call part I
A collection of various call part II
Yuyu's How to call mix video
Part 1 of Akiba Koala's Guide to Fan Chants
Part 1 of Akiba Koala's Guide to Fan Chants
PercyDW's guide to mixed call and fanchants (Shows Some Hello Project Styled Chants at a live)
A playlist o Idol Fancalls from the 80's (You better download it while it lasts)
https://www.youtube.com/playlist?list=PLVDYrLVksSzwB5nn0V0FBAQ5wRdofCn11