What is Degrees Of Freedom

"Degrees Of Freedom" or "DOF" is a number of axis and sensors combined for balancing a plane, a helicopter or a robot.

3 DOF : This could be a 3-axis accelerometer or it could be a 3-axis gyroscope.
6 DOF : This is mostly a 3-axis accelerometer combined with a 3-axis gyroscope.

Examples:
To control a remote control (RC) plane or helicopter or a self-balancing robot, both the information of the accelerometer and gyro are needed.
An other example of 6DOF is a combination of an accelerometer and a magnetometer for a tilt-compensated compass.
Many game controllers, phones and tablets contain a 6DOF sensor for motion information.

9 DOF : This is mostly a 6DOF, combined with a magnetometer (compass).
10 DOF : This could be a 9DOF, combined with a baromic pressure sensor. The baromic (or absolute pressure) sensor can be used as an indication for the height.
11 DOF : This could be the 10DOF, combined with a GPS module.

Multiple sensors in a chip or on a board.

A 9DOF module for example, could be a circuit board with three sensors. But some chips have a accelerometer and a gyro inside. In that case, only that sensor and a magnetometer is needed.

Interface with the Arduino

Some sensors have analog outputs, and can be easily connected to an Arduino. The software reads the analog signal with a single function: "analogRead()". Most Arduino's have only 6 analog inputs (the Arduino Mega has 16 analog inputs) so a 6DOF sensor can be connected without extra hardware to most Arduino boards.

Most sensors have an I2C or SPI interface. The analog-to-digital-conversion (ADC) is done inside the sensor. That increases the accuracy. Some sensors use 16-bits which is more than the 10-bits of the Arduino. Some sensor are very complicated and can be programmed with firmware.

To interface the Arduino with 6DOF, 9DOF, 10DOF, 11DOF sensors, see: http://playground.arduino.cc/Main/InterfacingWithHardware#DOF

Source: http://playground.arduino.cc/Main/WhatIsDegreesOfFreedom6DOF9DOF10DOF11DOF