The MPU-6050 is a 6DOF. In other words it has 6 Degrees Of Freedom, 3 axes of acceleration (x, y, z) and 3 axes of rotational movement (yaw, pitch, roll). It uses the I2C protocol.
The MPU-6050 is a gyroscope / accelerometer IC made by InvenSense. It can measure movement in 6 axes, 3 of acceleration and 3 of rotation. You can use it as an accelerometer and a gyroscope at the same time.
It uses the I2C protocol which makes it easy to combine this sensor with many other I2C using sensors.
Important: Notice that this IC runs on 3.3V which means 5V will burn it out. However, some breakouts, like the GY-521, have all that you need to connect an MPU-60X0 directly to your Arduino. The GY-521 has a voltage regulator so it's ok with 5V.
There is a good library on GitHub that is quite popular for this sensor. With detailed and understandable examples, beginners can use it too.