The simplest battery charger is a just a voltage source. When a battery is “dead” its voltage is lower than normal. When you attach the charger, since the charger’s voltage is higher, electric charge flows from it into the battery.
As the battery fills up with charge, the voltage rises. Eventually the voltage is the same as the charger and so no electric charge flows.
More complicated charges have a second stage that is trigged when the current drops to a certain level.
Smartphone chargers are actually built into the phone and do all sorts of fancy stuff while the phone is charging and most of that stuff is beyond the scope of EL5 in my opinion.
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