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There are two primary applications of the term Degrees of Freedom (DOF).

1) Denote the number and parameters of displacements and/or rotations available within a given task space.

Degrees of Freedom in a Three Dimensional (3-D) Space

In a Two Dimensional (2-D) space (like a table-top or the floor) there are three Degrees of Freedom. These include displacement along the X and Y axes, plus rotation.
In a Three Dimensional (3-D) space there are six degrees of freedom. These consist of displacement along three perpendicular axes (X, Y and Z), and rotation about those same axes.
DOF in 3D space are generally identified using the following nautical terms:
Displacements
Heave: Moving up and down
Surge: Moving forward and backward
Sway: Moving left and right

Rotations
Yaw: Turning left and right flight
Roll: Tilting side to side
Pitch: Tilting forward and backward>

2) Denote the number and parameters of displacements and/or rotations that a body or mechanism is able to execute.

Holonomic mechanisms are able to perform controlled movement through every available DOF within its particular task space.
Non-holonomic mechanisms are capable of fewer controlled DOF than are actually available within its task space.
Redundant mechanisms can execute more controllable DOF than are actually available within its task space.

[edit] Mobility In 2-D Space

Degrees of Freedom of a robot that utilizes Differential Steering

Degrees of Freedom of a robot that utilizes Ackerman Steering

A mobile robot that utilizes Ackerman (car-like) steering can theoretically reach any point and achieve any orientation within its 2-D task space, although it might have to carry out several and varied maneuvers to do so. This is due to the fact that this type of robot is only capable of executing a single displacement (forward/backward) and a single rotation (steering angle). This fact also means that this type of robot is capable of only two controlled DOF which is one fewer DOF than are available in its two dimensional task space. It therefore is non-holonomic.

A robot capable of Differential or Skid Steering would be able to move more directly from one point and orientation to a different point and orientation within the same 2-D task space. However it would still only be able to execute a single displacement (forward/backward) and a single rotation (turn, pivot or spin). Thus it can execute only two controlled DOF which again is one fewer than are available in its task space. So it too is non-holonomic.

A typical BEAM walker would also be non-holonomic because BEAM walkers can generally execute only a single displacement (forward/backward) and a single rotation (turning).

[edit] Holonomic Motion

Degrees of Freedom of a robot that utilizes Kiwi Drive

Degrees of Freedom of a robot that utilizes a 4 Wheel Omni-Drive

An example of a holonomic robot would be one that is equipped with three independently driven Mecanum wheels or omni-wheels, mounted in a triangular configuration. This is generally called a Kiwi Drive. A robot that utilizes Kiwi Drive can simultaneously execute controlled motion through both the X and Y axes, and rotational movement.

Holonomic Wheeled Robot Video #1

Holonomic Wheeled Robot Video #2

Holonomic Wheeled Robot Video #3

There are multi-legged walking robots (typically hexapods) that can execute displacement in both the X and Y axes, and rotation. These walkers are holonomic, but are not BEAM robots.

Holonomic Hexapod Video #1:

Holonomic Hexapod Video #2:

[edit] An Arm In 3-D Space

Layout of a Robotic Arm with Seven Degrees of Freedom

A normal human arm is redundant ( and therefore holonomic ) in that it has seven DOF. The shoulder gives pitch, yaw and roll. The elbow allows for pitch. The wrist allows for pitch and yaw. And the elbow and wrist together allow for Roll. Only three DOF are needed to move the hand to any particular point within a given three dimensional space, but having a greater the number of controlled DOF enables human beings to grasp items in that space from a variety of different angles and directions.

Robot arms are typically categorized by the number of controlled DOF they can execute. This number is equal to the sum of the DOF of each of a robot arm’s individual joints. Generally these will be either Hinge joints or Pivot joints, both of which are only capable of rotation about a single-axis. Therefore a robot arm that has seven individual Hinge joints or Pivot joints would be classified as a 7 DOF arm.

In some cases, complex joints capable of performing more than a single type of movement are used instead of the simpler hinge joints and pivot joints. Doing this increases the number of DOF that a robot arm would be able to execute, while keeping the number of actual joints to a minimum.

More information on various kinds of simple and complex joints can be found at: Types Of Joints.