A power supply converts the voltage coming out of the wall socket into a different voltage. A Power supply is needed to supply the charger with power. There are some chargers with built-in power supply, but those are the exception. 
The performance of the power supply influences the ability of the charger to charge your battery. If the supply is too weak, the charging process takes a lot of time or doesn't start at all. The upper limit depends on the ability of the charger and the charging capacity (C-Rate) of the battery pack.