How do animals and plants live in fast-flowing water?

Animals, such as fish, amphibians, mammals, and insects have a variety of adaptations that let them exist in swift moving water. 
Fast moving water tends to be colder than slow water; some of it originates as snow melt on the sides of mountains, for example. For this, animals in fast water tend to be more cold resistant. This is why you do not see fish like freshwater trout in equatorial regions. These animals struggle and die in slow, warm water.
Fast moving water means that if the animal or plant wants to stay put, it must resist the flow. Fish tend to be far more muscular and lean in these places, while insects like stoneflies, mayflies, and other larvae have ways to secure themselves to rocks and plants while growing. Plants develop strong root systems, and algae hold fast in rocky crevices in the river.
Some animals have to adapt their reproductive systems as well. Because finding a mate is difficult in the sweeping waters, breeding tends to happen all at one time. As an example, salmon have runs, where thousands of fish all swim upriver at once to breed, can dramatically alter the surrounding ecology of the river. Insects like mayflies have hatches, where hundreds of thousands of flies swarm into the air at once to breed and lay eggs.
Other adaptations include slow metabolisms, smaller overall sizes, and fast movement speed.
In the oceans, animals depend on currents in the water to live. Animals like jellyfish and plankton use currents to move around from one place to another, while others rely on currents to bring them nutrients they need to live. These currents also prevent the ocean from stagnating, and stir nutrients and gasses around to the same effect as stirring a compost heap.
Other animals, such as squid, whales, and even salmon, rely upon ocean currents to complete migration events for seasonal breeding, feeding, and other activities.
http://www.adfg.alaska.gov/index.cfm?adfg=wildlifenews.view_article&articles_id=407

Water flows at different speeds along different parts of a river's length.  The plants and animals that live in the fast-flowing stretches are specially adapted to life there.   Some cling to rocks or live under stones so that they are not swept away.