There are many types of kayak paddling just as there are many different types of kayaks.
The two videos below cover the most basic types of paddling, mostly applied to flat water kayaking. Of course, these strokes are every bit adaptable to flowing streams and rivers, but whitewater kayaking is almost a different sport requiring its own special blend of unique strokes and nuanced maneuvers.
These, however are the basics.
Recreational or fitness paddling with a flat bladed paddle
Fitness paddling or racing with a wing paddle
See also: The kayak wing paddle