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The most popular sensor chip is the MPU-6050 made by TDK - InvenSense. It is an incredibly cheap, very stable 6DoF chip, that is very basic and just combines a 3-axis gyroscope and a 3-axis accelerometer. It kicks out the raw data and you process it, externally. The chip is still popular today, although it is badly prone to drift. I found these to be solid and very reliable, if you ignore the drift.
The MPU-9250 (or 9255) is also made by TDK - InvenSense, and is billed as their second generation 9DoF chip, which "is a System in Package (SiP) that combines two chips: the MPU-6500, which contains a 3-axis gyroscope, a 3-axis accelerometer, and an onboard Digital Motion Processor™ (DMP™) capable of processing complex MotionFusion algorithms; and the AK8963, the market leading 3-axis digital compass". Like the MPU-6050, the chip kicks out raw data for external processing. The AK8963 (magnetometer) allows for drift compensation through complex calculations. Note: InvenSense have obsoleted this chip & are advising that it not be used for new developments. My own experiences of these chips is that they are very difficult to get reliable results from and I wouln't now choose to use them for anything.
Then came a game-changer. Bosch Sensortec released the BNO055, which alongside the the now standard 9DoF gyroscope, accelerometer and magnetometer has its own built in "processor". This means that it can calculate its own quaternions, without any external calculations. This simplified everything, and seems to have widely been accepted as the way forwards. The customer reaction to the BNO055 has been quite mixed. Many people complained that it didn't do what the needed, or what it should have. Many others really loved it. I found my own to be excellent, and that it does everything that I could ask of it in a very simple to use package. Not long after it was released though, it was superseded by the BNO080.
Bosch Sensortec, in cooperation with Hillcrest, released the BNO080, which "features a high-performance accelerometer, magnetometer, and gyroscope with a low-power 32-bit ARM Cortex M0+ MCU". This offered everything that the BNO055 promised, only faster and with lower power requirements. Apparently, many of the people who were unhappy with the BNO055 upgraded to the BNO080 and were really very happy with their it. The BNO080 seems to have excellent user reviews, although I haven't tried one myself yet.