From the USGS:
Stream stage is an important concept when analyzing how much water is moving in a stream at any given moment. “Stage” is the water level above some arbitrary point in the river and is commonly measured in feet. For example, on a normal day when no rain has fallen for a while, a river might have a stage of 2 feet. If a big storm hits, the river stage could rise to 15 or 20 feet, sometimes very quickly. This is important because past records might tell us that when the stage hits 21 feet, the water will start flowing over its banks and into the basements of houses along the river — time to tell those people to move out! With modern technology, the USGS can monitor the stage of many streams almost instantly.
Hydrologists are able to convert stage height into streamflow volume by determining a rating curve for each site.
Learn more:
Streamgaging Basics National Water Information System (NWIS) Mapper