In general, the current flows to charge a battery because the charger voltage is higher than the battery voltage. As the voltage rises the difference in voltages becomes less so the current drops. In theory, when the two are equal no current flows at all. In the real World, the charge voltage is always higher than the maximum the battery can reach and other measures, such as current limiting or timed cut-off are used to ensure it doesn't get over charged.
Laptops are a special case as the manufacturers push the batery to their limits to make the charging faster, usually the battery pack has a voltage and temperature monitor inside it to maximize the charge rate without risking damage. At full charge, the monitor switches to trickle charge mode to keep the battery "topped up".
As for conserving energy, if the cell is not demanding charge, no current flows so the charger output is present but not being used.
Brian.