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All other things equal in quadcopters (motors kv, blades size-pitch), higher voltage = greater lift from motors. I am not saying that voltage is the only factor to affect lift, but it is decisive. Have you seen many mini quads with 3s batteries?
2s, 3s, 4s etc refers to the battery voltage.1s is a single cell Lipo battery (3,6 Volt),2s is a two cell LiPo battery (2x3,6V= 7,2 Volt),3s is a three cell Lipo battery (10,8V) and so on.Obviously, the higher the voltage, the greater the lift capacity of the quadcopter.