"Wait!" you ask, "Why do you need flight controller for a simulator?". The primary job of flight controller is to take in desired state as input, estimate actual state using sensors data and then drive the actuators in such a way so that actual state comes as close to the desired state. For quadrotors, desired state can be specified as roll, pitch and yaw, for example. It then estimates actual roll, pitch and yaw using gyroscope and accelerometer. Then it generates appropriate motor signals so actual state becomes desired state. You can find more in-depth in our paper.
Simulator consumes the motor signals generated by flight controller to figure out force and thrust generated by each actuator (i.e. propellers in case of quadrotor). This is then used by the physics engine to compute the kinetic properties of the vehicle. This in turn generates simulated sensor data and feed it back to the flight controller. You can find more in-depth in our paper.
Hardware-in-Loop (HITL or HIL) means flight controller runs in actual hardware such as Naze32 or Pixhawk chip. You then connect this hardware to PC using USB port. Simulator talks to the device to retrieve actuator signals and send it simulated sensor data. This is obviously as close as you can get to real thing. However, it typically requires more steps to set up and usually hard to debug. One big issue is that simulator clock and device clock runs on their own speed and accuracy. Also, USB connection (which is usually only USB 2.0) may not be enough for real-time communication.
In "software-in-loop" simulation (SITL or SIL) mode the firmware runs in your computer as opposed to separate board. This is generally fine except that now you are not touching any code paths that are specific to your device. Also, none of your code now runs with real-time clock usually provided by specialized hardware board. For well-designed flight controllers with software clock, these are usually not concerning issues.
AirSim has built-in flight controller called simple_flight and it is used by default. You don't need to do anything to use or configure it. AirSim also supports PX4 & ArduPilot as external flight controllers for advanced users.
Yes, now it's possible to use AirSim without flight controller. Please see the instructions here for how to use so-called "Computer Vision" mode. If you don't need vehicle dynamics, we highly recommend using this mode.