Hi Michael welcome to the forum! LitterBug is right about APM vs. Pixhawk. The APM hardware has been succeeded by the Pixhawk, and even the Pixhawk has been succeeded by the Pixhawk 2. But the Pixhawk is still better for hobbyists as the Pixhawk 2 is meant for more industrial applications. The APM hardware is limited and does not support the latest arducopter (the popular firmware used) releases. However the APM is still very capable and can often be found for cheap.
This next bit applies to both APM/Pixhawk should you get one or the other. Typically if you buy the 'full' kit you get the flight controller, a GPS/Compass module, a current/voltage power module, an external buzzer, and a LED button. Some places only sell the flight controller and you can find the other parts yourself. Both hardwares can run the most popular firmware, arducopter. Arducopter is flashed and configured through Mission Planner on your computer. Other firmwares exist, but arducopter with mission planner is the friendliest and most intuitive to use (at least on the surface).
Here are some peripherals and their descriptions:
The GPS is needed for all GPS based functions like return to home (RTH), position hold, flying waypoints, etc. Some GPS modules are more accurate than others. A really popular one that works well is the UBLOX M8N GPS.
The power module reports voltage and current data to the flight controller for use in low battery detection. The flight controller also takes this data (and other internal sensor data) and stores it on a micro SD card so you can view details about each flight later on your computer.
The external buzzer is to alert you of various warnings or setting changes like low battery, poor compass health, or just to confirm arming the copter. The button is an extra safety mechanism to prevent accidental arming. You have to press down the button to initialize the ESCs and before you can proceed to arm via the transmitter.
An optional peripheral that may or may not come with the package is a telemetry radio module. One radio connects to the flight controller on the multirotor and the other plugs in to your computer via USB. If you are using Ardupilot, the program you interface with on your computer is called Mission Planner. The telemetry radios allow you to get data in flight from the flight controller and even allow for you to control and change settings on the flying multirotor right from you computer. A bluetooth module is another option that does the same thing as a telemetry radio, but has much less range.
Another popular extra peripheral that is rarely found in the full package is an on screen display (OSD). The OSD takes important data like battery voltage, altitude, direction, etc, and encodes your FPV video stream to have this information when you look at your FPV screen/goggles. If you are not doing FPV you will have no use for an OSD. Plus all the information on the OSD can be found on the computer if you use a telemetry radio.
If you are an expert, which obviously you are not since you are new, there are numerous other peripherals you can add. You won't be able to use all these at once, but the Pixhawk will be able to run more than the APM. You could add more GPS modules to provide redundancy, you could add an ultrasonic sensor to have more accurate low altitude hovering. There is also optical flow sensors, LIDAR modules, parachutes, landing gear, and more which you can read about on ardupilot.org which LitterBug linked to. As you can see the APM and Pixhawk hardwares are design to work from a beginner level, all the way to an expert and even industrial level.
