One milliampere-hour is exactly 0.001 ampere-hours. Both measure electric charge, and both share a limitation worth stating immediately: neither tells you how much energy a battery holds without the voltage.
About the units
An ampere-hour is the charge transferred by a current of one ampere flowing for one hour. It is a charge, not an energy — to get energy you multiply by the voltage, which is why watt-hours rather than ampere-hours appear in airline battery regulations. The milliampere-hour became the consumer standard because phone and laptop cells fall naturally in the range of a few thousand of them. A 5,000 mAh power bank at 3.7 V holds 18.5 Wh; the same 5,000 mAh at 12 V would hold three times as much, and the label on the box does not distinguish them.
The exact factor
The factor is exactly 1,000. Reference values: an AA alkaline cell is about 2,500 mAh at 1.5 V; a smartphone battery 4,000 to 5,000 mAh at 3.85 V; a laptop battery 5,000 to 8,000 mAh at 11.1 V; and an electric car pack 100 to 200 Ah at 350 to 800 V. The last figure is why electric vehicles are specified in kilowatt-hours instead — at pack level the ampere-hour figure alone is meaningless without the system voltage.
Where you meet this conversion
Battery specification and comparison across every kind of device. Power bank marketing uses milliampere-hours because the numbers are large and impressive, which is also why comparing two power banks by mAh alone is unreliable: the internal cell voltage and the conversion losses in the USB output both matter. Drone and radio-control battery packs are labelled in mAh together with a cell count that gives the voltage. Battery testers and chargers report capacity in the same unit.