A river begins at a source (or more often several sources) which is usually a watershed, drains all the streams in its drainage basin, follows a path called a rivercourse (or just course) and ends at either at a mouth or mouths which could be a confluence, river delta, etc.

Where does a river start?

The place where a river begins is called its source. River sources are also called headwaters. Rivers often get their water from many tributaries, or smaller streams, that join together. The tributary that started the farthest distance from the river's end would be considered the source, or headwaters.

Where does a river end?

Eventually a river meets the sea and the place where it does is called the mouth. The last of the mud is deposited at the river's mouth. A wide mouth is called an estuary.