A watershed is an area of land over which rain and other sources of water drain into a common body of water, such as a river or lake. If you could draw a line from hilltop to hilltop around a river and its tributaries (which are the streams that feed into the river), you would be outlining its watershed.
Any rain that falls on the river-side of the edge of the watershed could flow downhill into the river or one of its tributary streams. Water on the other side of the dotted line flows away and is part of a different watershed.
Anything that happens in the watershed can potentially affect the quality of a body of water.
If you found pollution in the water at a particular point and you have mapped the watershed, you would know where to look for the source of the pollution; that is, upstream, within the watershed, rather than downstream or outside the watershed.
Point sources of pollution, such as a discharge location at a factory, are often easier to remedy. You can identify the source and make a change.
Nonpoint sources of pollution are harder to identify, as they come from many diffuse sources, including air pollution and runoff from agricultural land. As rainfall or snowmelt flows over the land, it picks up pollutants and flows with them into the bodies of water in the watershed.