An ampere is a coulomb/second. A coulomb is a large number of charge (6.24×10^18 ). So an amp is a rate of flow of charge (typically electrons), that specifies how many coulombs move every second. An ampere•hour is simply how long you can maintain that current. So 1ampere•hour means you can sustain a current of 1 amp for one hour. So that means you have moved 3600 coulombs of charge in that time.
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