"Degrees Of Freedom" or "DOF" is a number of axis and sensors combined for balancing a plane, a helicopter or a robot.
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.
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
Wikipedia about Degrees Of Freedom: http://en.wikipedia.org/wiki/Degrees_of_freedom_%28mechanics%29