A holonomic drive can instantaneously move in any direction. It does not need to do any complex motions to achieve a particular heading. This type of robot would have 2 degrees of freedom in that it can move in both the X and Y plane freely. Simply put it is omni directional. A holonomic drive is useful for situations requiring higher mobility and lower traction than a standard drive system. Using driver oriented control, joystick mixing, and drive mixing, it requires much less thought to adjusting for various conditions on the part of the driver.
Non-holonomic drive can not instantaneously move in any direction. An example of a non-holonomic drive would be a car. Usually a non-holonomic drive uses Ackerman steering; again, such as a car.
Typically a holonomic drive uses 3 or 4 omni wheels. Omniwheels are wheels that can move in any direction, at any angle, without rotating beforehand.
A 3 wheel configuration would be a kiwi drive. A kiwi drive is 3 omni wheels each separated by 120 degrees each. Typically a kiwi drive is in a triangular configuration. All 3 omni wheels are independently driven.
Sources:
Society of Robots
BEAM Wiki
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