A unit for measuring electric power over time. mAh is commonly used to describe the total amount of energy a battery can store at one time.
A battery rated for more mAh will power a phone for a longer amount of time, given the same usage pattern. The trade-off is that batteries with more mAh are generally also physically larger and heavier.
More technically, a higher mAh rating means the (fully-charged) battery can power a device that consumes more power and/or for a longer amount of time before becoming depleted and needing to be re-charged. For example, a battery rated at 1500 mAh can power a device drawing 100 milliamps for 15 hours, or a device drawing 150 milliamps for 10 hours. (In other words, a device using more power will drain the same battery faster.) In that same example, a larger battery, rated 3000 mAh, could power a device drawing 100 milliamps for 30 hours.