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.
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.
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. There seemed to be a degree of dissatisfaction amongst some users of the BNO055, and it was soon superseded.
Next, 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 people who were unhappy with the BNO055 upgraded to the BNO080 and were really very happy with their it. The BNO055 seems to have excellent user reviews