The Full form of MAH is Milliampere Hours. MAH is a unit for measuring electric power of a battery over time. MAH is usually used to describe the total amount of energy a battery can store at one time. A battery with more MAH will power an electronic device for a longer duration, given the same usage pattern. The trade-off is that batteries rated for more MAH are generally also physically heavier and larger.
Technically, a higher MAH rating means the battery that is fully-charged can power an electronic device that consumes more power and/or for a longer period of time before needing to be re-charged or becoming depleted. For example, a phone battery rated at 1400 mAh can power a phone drawing 100 milliamps for 14 hours, or a phone drawing 140 milliamps for 10 hours. In other words, a device consuming more power will deplete the same battery faster. In the same example, a larger battery, rated 2800 mAh, could power a device drawing 100 milliamps for 28 hours or drawing 100 milliamps for 28 hours.