Salmon Run can be thought of as Alley Salmon - like the same programmer's Alley Cat you play an animal ultimately seeking a mate, although this interaction is not part of the gameplay. As a salmon, you must swim (at variable speed) up a river full of jagged edges, avoiding too much contact with these.
Other hazards include waterfalls (you must jump over these), bears (dive underwater to prevent them spotting you), birds (dart around them by adjusting your speed) and fishermen (stay away from the edges to avoid being entangled). Points are awarded based on how far upstream you can progress. You start with only one life, but completing a level grants you an extra life, a points bonus for any time remaining, and a short romantic interlude animation.