Water is always in motion, but generally the current strengths decrease with depth. On the abyssal plains of the vast ocean baisns, the motion is very slow, but along the ocean margins and the mid-ocean ridges currents can be rather stronger. Around islands and seamounts currents can accellerate and form eddies and outher special features.
Surface currents are wind-driven, but other factors such as tides and heat exchange are also important. The rotation of the Earth also strongly influences current patterns.
On www.mare-eco.no/sci there is a topic on environmental issues that has an overview article with more in-depth information.